mmtheorems22 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 2101-2200 - Page 22 of 108

Type	Label	Description
Statement
Theorem	eqsstrd 2101	Substitution of equality into a subclass relationship.
|- (ph -> A = B)   &   |- (ph -> B (_ C)   =>   |- (ph -> A (_ C)
Theorem	eqsstr3d 2102	Substitution of equality into a subclass relationship.
|- (ph -> B = A)   &   |- (ph -> B (_ C)   =>   |- (ph -> A (_ C)
Theorem	sseqtrd 2103	Substitution of equality into a subclass relationship.
|- (ph -> A (_ B)   &   |- (ph -> B = C)   =>   |- (ph -> A (_ C)
Theorem	sseqtr4d 2104	Substitution of equality into a subclass relationship.
|- (ph -> A (_ B)   &   |- (ph -> C = B)   =>   |- (ph -> A (_ C)
Theorem	3sstr3 2105	Substitution of equality in both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- A (_ B   &   |- A = C   &   |- B = D   =>   |- C (_ D
Theorem	3sstr4 2106	Substitution of equality in both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- A (_ B   &   |- C = A   &   |- D = B   =>   |- C (_ D
Theorem	3sstr3g 2107	Substitution of equality into both sides of a subclass relationship.
|- (ph -> A (_ B)   &   |- A = C   &   |- B = D   =>   |- (ph -> C (_ D)
Theorem	3sstr4g 2108	Substitution of equality into both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- (ph -> A (_ B)   &   |- C = A   &   |- D = B   =>   |- (ph -> C (_ D)
Theorem	3sstr3d 2109	Substitution of equality into both sides of a subclass relationship.
|- (ph -> A (_ B)   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> C (_ D)
Theorem	3sstr4d 2110	Substitution of equality into both sides of a subclass relationship. (The proof was shortened by Eric Schmidt, 26-Jan-2007.)
|- (ph -> A (_ B)   &   |- (ph -> C = A)   &   |- (ph -> D = B)   =>   |- (ph -> C (_ D)
Theorem	syl5ss 2111	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- C = A   =>   |- (ph -> C (_ B)
Theorem	syl5ssr 2112	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- A = C   =>   |- (ph -> C (_ B)
Theorem	syl6ss 2113	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- B = C   =>   |- (ph -> A (_ C)
Theorem	syl6ssr 2114	A chained subclass and equality deduction.
|- (ph -> A (_ B)   &   |- C = B   =>   |- (ph -> A (_ C)
Theorem	eqimss 2115	Equality implies the subclass relation.
|- (A = B -> A (_ B)
Theorem	eqimss2 2116	Equality implies the subclass relation.
|- (B = A -> A (_ B)
Theorem	eqimssi 2117	Infer subclass relationship from equality.
|- A = B   =>   |- A (_ B
Theorem	eqimss2i 2118	Infer subclass relationship from equality.
|- A = B   =>   |- B (_ A
Theorem	nss 2119	Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18.
|- (-. A (_ B <-> E.x(x e. A /\ -. x e. B))
Theorem	ssralv 2120	Quantification restricted to a subclass.
|- (A (_ B -> (A.x e. B ph -> A.x e. A ph))
Theorem	ssrexv 2121	Existential quantification restricted to a subclass.
|- (A (_ B -> (E.x e. A ph -> E.x e. B ph))
Theorem	ss2ab 2122	Class abstractions in a subclass relationship.
|- ({x | ph} (_ {x | ps} <-> A.x(ph -> ps))
Theorem	abss 2123	Class abstraction in a subclass relationship.
|- ({x | ph} (_ A <-> A.x(ph -> x e. A))
Theorem	ssab 2124	Subclass of a class abstraction.
|- (A (_ {x | ph} <-> A.x(x e. A -> ph))
Theorem	ssabral 2125	The relation for a subclass of a class abstraction is equivalent to restricted quantification.
|- (A (_ {x | ph} <-> A.x e. A ph)
Theorem	ss2abi 2126	Inference of abstraction subclass from implication.
|- (ph -> ps)   =>   |- {x | ph} (_ {x | ps}
Theorem	abssdv 2127	Deduction of abstraction subclass from implication.
|- (ph -> (ps -> x e. A))   =>   |- (ph -> {x | ps} (_ A)
Theorem	abssi 2128	Inference of abstraction subclass from implication.
|- (ph -> x e. A)   =>   |- {x | ph} (_ A
Theorem	ss2rab 2129	Restricted abstraction classes in a subclass relationship.
|- ({x e. A | ph} (_ {x e. A | ps} <-> A.x e. A (ph -> ps))
Theorem	rabss 2130	Restricted class abstraction in a subclass relationship.
|- ({x e. A | ph} (_ B <-> A.x e. A (ph -> x e. B))
Theorem	ssrab 2131	Subclass of a restricted class abstraction.
|- (B (_ {x e. A | ph} <-> (B (_ A /\ A.x e. B ph))
Theorem	ssrabdv 2132	Subclass of a restricted class abstraction (deduction rule).
|- (ph -> B (_ A)   &   |- ((ph /\ x e. B) -> ps)   =>   |- (ph -> B (_ {x e. A | ps})
Theorem	ss2rabdv 2133	Deduction of restricted abstraction subclass from implication.
|- ((ph /\ x e. A) -> (ps -> ch))   =>   |- (ph -> {x e. A | ps} (_ {x e. A | ch})
Theorem	ss2rabi 2134	Inference of restricted abstraction subclass from implication.
|- (x e. A -> (ph -> ps))   =>   |- {x e. A | ph} (_ {x e. A | ps}
Theorem	rabss2 2135	Subclass law for restricted abstraction.
|- (A (_ B -> {x e. A | ph} (_ {x e. B | ph})
Theorem	ssab2 2136	Subclass relation for the restriction of a class abstraction.
|- {x | (x e. A /\ ph)} (_ A
Theorem	ssrab2 2137	Subclass relation for a restricted class.
|- {x e. A | ph} (_ A
Theorem	uniiunlem 2138	A subset relationship useful for converting union to indexed union using dfiun2 2594 or dfiun2g 2593 and intersection to indexed intersection using dfiin2 2595.
|- (A.x e. A B e. D -> (A.x e. A B e. C <-> {y | E.x e. A y = B} (_ C))
Theorem	dfpss2 2139	Alternate definition of proper subclass.
|- (A (. B <-> (A (_ B /\ -. A = B))
Theorem	dfpss3 2140	Alternate definition of proper subclass.
|- (A (. B <-> (A (_ B /\ -. B (_ A))
Theorem	psseq1 2141	Equality theorem for proper subclass.
|- (A = B -> (A (. C <-> B (. C))
Theorem	psseq2 2142	Equality theorem for proper subclass.
|- (A = B -> (C (. A <-> C (. B))
Theorem	psseq1i 2143	An equality inference for the proper subclass relationship.
|- A = B   =>   |- (A (. C <-> B (. C)
Theorem	psseq2i 2144	An equality inference for the proper subclass relationship.
|- A = B   =>   |- (C (. A <-> C (. B)
Theorem	psseq12i 2145	An equality inference for the proper subclass relationship.
|- A = B   &   |- C = D   =>   |- (A (. C <-> B (. D)
Theorem	psseq1d 2146	An equality deduction for the proper subclass relationship.
|- (ph -> A = B)   =>   |- (ph -> (A (. C <-> B (. C))
Theorem	psseq2d 2147	An equality deduction for the proper subclass relationship.
|- (ph -> A = B)   =>   |- (ph -> (C (. A <-> C (. B))
Theorem	psseq12d 2148	An equality deduction for the proper subclass relationship.
|- (ph -> A = B)   &   |- (ph -> C = D)   =>   |- (ph -> (A (. C <-> B (. D))
Theorem	pssss 2149	A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23.
|- (A (. B -> A (_ B)
Theorem	pssssd 2150	Deduce subclass from proper subclass.
|- (ph -> A (. B)   =>   |- (ph -> A (_ B)
Theorem	sspss 2151	Subclass in terms of proper subclass.
|- (A (_ B <-> (A (. B \/ A = B))
Theorem	pssirr 2152	Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23.
|- -. A (. A
Theorem	pssn2lp 2153	Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23.
|- -. (A (. B /\ B (. A)
Theorem	sspsstri 2154	Two ways of stating trichotomy with respect to inclusion.
|- ((A (_ B \/ B (_ A) <-> (A (. B \/ A = B \/ B (. A))
Theorem	ssnpss 2155	Partial trichotomy law for subclasses.
|- (A (_ B -> -. B (. A)
Theorem	psstr 2156	Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23.
|- ((A (. B /\ B (. C) -> A (. C)
Theorem	sspsstr 2157	Transitive law for subclass and proper subclass.
|- ((A (_ B /\ B (. C) -> A (. C)
Theorem	psssstr 2158	Transitive law for subclass and proper subclass.
|- ((A (. B /\ B (_ C) -> A (. C)
The difference, union, and intersection of two classes
Theorem	difeq1 2159	Equality theorem for class difference.
|- (A = B -> (A \ C) = (B \ C))
Theorem	difeq2 2160	Equality theorem for class difference.
|- (A = B -> (C \ A) = (C \ B))
Theorem	difeq1i 2161	Inference adding difference to the right in a class equality.
|- A = B   =>   |- (A \ C) = (B \ C)
Theorem	difeq2i 2162	Inference adding difference to the left in a class equality.
|- A = B   =>   |- (C \ A) = (C \ B)
Theorem	difeq12i 2163	Equality inference for class difference.
|- A = B   &   |- C = D   =>   |- (A \ C) = (B \ D)
Theorem	difeq1d 2164	Deduction adding difference to the right in a class equality.
|- (ph -> A = B)   =>   |- (ph -> (A \ C) = (B \ C))
Theorem	difeq2d 2165	Deduction adding difference to the left in a class equality.
|- (ph -> A = B)   =>   |- (ph -> (C \ A) = (C \ B))
Theorem	difeqri 2166	Inference from membership to difference.
|- ((x e. A /\ -. x e. B) <-> x e. C)   =>   |- (A \ B) = C
Theorem	hbdif 2167	Bound-variable hypothesis builder for class difference.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A \ B) -> A.x y e. (A \ B))
Theorem	eldifi 2168	Implication of membership in a class difference.
|- (A e. (B \ C) -> A e. B)
Theorem	eldifn 2169	Implication of membership in a class difference.
|- (A e. (B \ C) -> -. A e. C)
Theorem	elndif 2170	A set does not belong to a class excluding it.
|- (A e. B -> -. A e. (C \ B))
Theorem	neldif 2171	Implication of membership in a class difference.
|- ((