mmtheorems46 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4501-4600 - Page 46 of 108

Type	Label	Description
Statement
Theorem	mapdom1 4501	Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ~<_ B -> (A ^m C) ~<_ (B ^m C))
Theorem	mapdom2lem 4502	Lemma for mapdom2 4503.
Theorem	mapdom2 4503	Order-preserving property of set exponentiation. Theorem 6L(d) of [Enderton] p. 149.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A ~<_ B /\ -. (A = (/) /\ C = (/))) -> (C ^m A) ~<_ (C ^m B))
Theorem	mapxpen 4504	Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A ^m B) ^m C) ~~ (A ^m (B X. C))
Theorem	xpmapenlem1 4505	Lemma for xpmapen 4510.
Theorem	xpmapenlem2 4506	Lemma for xpmapen 4510.
Theorem	xpmapenlem3 4507	Lemma for xpmapen 4510.
Theorem	xpmapenlem4 4508	Lemma for xpmapen 4510.
Theorem	xpmapenlem5 4509	Lemma for xpmapen 4510.
Theorem	xpmapen 4510	Equinumerosity law for set exponentiation of a cross product. Exercise 4.47 of [Mendelson] p. 255.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A X. B) ^m C) ~~ ((A ^m C) X. (B ^m C))
Theorem	mapunen 4511	Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A i^i B) = (/) -> (C ^m (A u. B)) ~~ ((C ^m A) X. (C ^m B)))
Theorem	pwen 4512	If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87.
|- B e. V   =>   |- (A ~~ B -> P~A ~~ P~B)
Theorem	ssenen 4513	Equinumerosity of equinumerous subsets of a set.
|- A e. V   &   |- B e. V   =>   |- (A ~~ B -> {x | (x (_ A /\ x ~~ C)} ~~ {x | (x (_ B /\ x ~~ C)})
Theorem	limenpsi 4514	A limit ordinal is equinumerous to a proper subset of itself.
|- Lim A   =>   |- (A e. B -> A ~~ (A \ {(/)}))
Theorem	limensuci 4515	A limit ordinal is equinumerous to its successor.
|- Lim A   =>   |- (A e. B -> A ~~ suc A)
Theorem	limensuc 4516	A limit ordinal is equinumerous to its successor.
|- ((A e. B /\ Lim A) -> A ~~ suc A)
Pigeonhole Principle
Theorem	phplem1 4517	Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element.
Theorem	phplem2 4518	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.
Theorem	phplem3 4519	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.
Theorem	phplem4 4520	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.
Theorem	nneneq 4521	Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136.
|- ((A e. om /\ B e. om) -> (A ~~ B <-> A = B))
Theorem	php 4522	Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 4517 through phplem4 4520, nneneq 4521, and this final piece of the proof.
|- ((A e. om /\ B (. A) -> -. A ~~ B)
Theorem	php2 4523	Corollary of Pigeonhole Principle.
|- ((A e. om /\ B (. A) -> B ~< A)
Theorem	php3 4524	Corollary of Pigeonhole Principle. If A is finite and B is a proper subset of A, the B is strictly less numerous than A. Stronger version of Corollary 6C of [Enderton] p. 135.
|- ((A e. Fin /\ B (. A) -> B ~< A)
Theorem	php3OLD 4525	Corollary of Pigeonhole Principle. If A is finite and B is a proper subset of A, the B is strictly less numerous than A. Stronger version of Corollary 6C of [Enderton] p. 135. (The expression E.x e. omA ~~ x is the definition of "finite," and "infinite" is defined as "not finite.")
|- ((E.x e. om A ~~ x /\ B (. A) -> B ~< A)
Theorem	php4 4526	Corollary of the Pigeonhole Principle php 4522: a natural number is strictly dominated by its successor.
|- (A e. om -> A ~< suc A)
Theorem	php5 4527	Corollary of the Pigeonhole Principle php 4522: a natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90.
|- (A e. om -> -. A ~~ suc A)
Finite sets
Theorem	onomeneq 4528	An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse.
|- ((A e. On /\ B e. om) -> (A ~~ B <-> A = B))
Theorem	onfin 4529	An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92.
|- (A e. On -> (A e. Fin <-> A e. om))
Theorem	nndomo 4530	Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146.
|- ((A e. om /\ B e. om) -> (A ~<_ B <-> A (_ B))
Theorem	nnsdomo 4531	Cardinal ordering agrees with natural number ordering.
|- ((A e. om /\ B e. om) -> (A ~< B <-> A (. B))
Theorem	omsucdom 4532	Strict dominance of natural numbers is the same as dominance over the successor of the smaller.
|- ((A e. om /\ B e. om) -> (A ~< B <-> suc A ~<_ B))
Theorem	sucdomi 4533	Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 4864.
|- ((A e. om /\ B e. C) -> (suc A ~<_ B -> A ~< B))
Theorem	0sdom1dom 4534	Strict dominance over zero is the same as dominance over one.
|- A e. V   =>   |- ((/) ~< A <-> 1o ~<_ A)
Theorem	1sdom2 4535	Ordinal 1 is strictly dominated by ordinal 2.
|- 1o ~< 2o
Theorem	finsucdom 4536	Strict dominance of a finite set over a natural number is the same as dominance over its successor.
|- ((A e. om /\ B e. Fin) -> (A ~< B <-> suc A ~<_ B))
Theorem	finsucdomOLD 4537	Strict dominance of a finite set over a natural number is the same as dominance over its successor.
|- ((A e. om /\ E.x e. om B ~~ x) -> (A ~< B <-> suc A ~<_ B))
Theorem	pssinf 4538	A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136.
|- ((A (. B /\ A ~~ B) -> -. B e. Fin)
Theorem	pssinfOLD 4539	A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136.
|- ((A (. B /\ A ~~ B) -> -. E.x e. om B ~~ x)
Theorem	ominf 4540	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.
|- -. om e. Fin
Theorem	ominfOLD 4541	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.
|- -. E.x e. om om ~~ x
Theorem	omsdomnn 4542	Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. Here we use A ~<_ om /\ -. om ~~ A instead of A ~< om because, due to a peculiarity ultimately caused our ordered pair definition, we would need the Axiom of infinity (which we have avoided up to now) in order to prove the latter.
|- (A e. om -> (A ~<_ om /\ -. om ~~ A))
Theorem	isfinite1 4543	Omega strictly dominates a finite set. See comment in omsdomnn 4542.
|- (A e. Fin -> (A ~<_ om /\ -. om ~~ A))
Theorem	isfinite1OLD 4544	Omega strictly dominates a finite set. See comment in omsdomnn 4542.
|- (E.x e. om A ~~ x -> (A ~<_ om /\ -. om ~~ A))
Theorem	infsdomnn 4545	An infinite set strictly dominates a natural number.
|- A e. V   =>   |- ((om ~<_ A /\ B e. om) -> B ~< A)
Theorem	infn0 4546	An infinite set is not empty.
|- A e. V   =>   |- (om ~<_ A -> A =/= (/))
Theorem	enfi 4547	Equinmerous sets have the same finiteness.
|- ((B e. C /\ A ~~ B) -> (A e. Fin <-> B e. Fin))
Theorem	pssnn 4548	A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137.
|- ((A e. om /\ B (. A) -> E.x e. A B ~~ x)
Theorem	ssnnfi 4549	A subset of a natural number is finite.
|- ((A e. om /\ B (_ A) -> B e. Fin)
Theorem	ssnnfiOLD 4550	A subset of a natural number is finite.
|- ((A e. om /\ B (_ A) -> E.x e. om B ~~ x)
Theorem	ssfi 4551	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((A e. Fin /\ B (_ A) -> B e. Fin)
Theorem	ssfiOLD 4552	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((E.x e. om A ~~ x /\ B (_ A) -> E.x e. om B ~~ x)
Theorem	domfi 4553	A set dominated by a finite set is finite.
|- ((A e. Fin /\ B ~<_ A) -> B e. Fin)
Theorem	domfiOLD 4554	A set dominated by a finite set is finite.
|- ((E.x e. om A ~~ x /\ B ~<_ A) -> E.x e. om B ~~ x)
Theorem	unblem1 4555	Lemma for unbnn 4559. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set.
Theorem	unblem2 4556	Lemma for unbnn 4559. The value of the function F belongs to the unbounded set of natural numbers A.
Theorem	unblem3 4557	Lemma for unbnn 4559. The value of the function F is less than its value at a successor.
Theorem	unblem4 4558	Lemma for unbnn 4559. The function F maps the set of natural numbers one-to-one to the set of unbounded natural numbers A.
Theorem	unbnn 4559	Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnnt 4661 for a stronger version without the hypothesis.
|- A e. V   =>   |- ((A (_ om /\ A.x e. om E.y e. A x e. y) -> A ~~ om)
Theorem	unbnn2 4560	Version of unbnn 4559 that does not require a strict upper bound.
|- A e. V   =>   |- ((A (_ om /\ A.x e. om E.y e. A x (_ y) -> A ~~ om)
Theorem	isfinite2 4561	Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity.
|- (A ~< om -> A e. Fin)
Theorem	isfinite2OLD 4562	Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity.
|- (A ~< om -> E.x e. om A ~~ x)
Theorem	fin2inf 4563	This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of strict dominance, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless om exists.
|- (A ~< om -> om e. V)
Theorem	unfilem1 4564	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem2 4565	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem3 4566	Lemma for proving that the union of two finite sets is finite.
Theorem	unfi 4567	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144.
|- ((A e. Fin /\ B e. Fin) -> (A u. B) e. Fin)
Theorem	unfiOLD 4568	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144.
|- ((E.x e. om A ~~ x /\ E.x e. om B ~~ x) -> E.x e. om (A u. B) ~~ x)
Theorem	unfi2 4569	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 4567 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4563).
|- ((A ~< om /\ B ~< om) -> (A u. B) ~< om)
Theorem	unfi2OLD 4570	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 4567 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4563).
|- ((A ~< om /\ B ~< om) -> (A u. B) ~< om)
Theorem	infcntss 4571	Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.)
|- A e. V   =>   |- (om ~<_ A -> E.x(x (_ A /\ x ~~ om))
Theorem	prfi 4572	An unordered pair is finite.
|- {A, B} e. Fin
Theorem	prfiOLD 4573	An unordered pair is finite.
|- E.x e. om {A, B} ~~ x
Theorem	unifi 4574	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144.
|- ((A e. Fin /\ A.x e. A x e. Fin) -> U.A e. Fin)
Theorem	unifi2 4575	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 4574 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4563).
|- ((A ~< om /\ A.x e. A x ~< om) -> U.A ~< om)
Theorem	unifiOLD 4576	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144.
|- ((E.n e. om A ~~ n /\ A.x e. A E.n e. om x ~~ n) -> E.n e. om U.A ~~ n)
Theorem	unifi2OLD 4577	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 4574 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4563).
|- ((A ~< om /\ A.x e. A x ~< om) -> U.A ~< om)
Theorem	fiint 4578	Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of A is in A." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally.
|- (A.x e. A A.y e. A (x i^i y) e. A <-> A.x((x (_ A /\ x =/= (/) /\