Theorem find 3162

Description: The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that A is a set of natural numbers, zero belongs to A, and given any member of A the member's successor also belongs to A. The conclusion is that every natural number is in A.

Hypothesis

Ref	Expression
find.1	|- (A (_ om /\ (/) e. A /\ A.x e. A suc x e. A)

Assertion

Ref	Expression
find	|- A = om

Distinct variable group:   x,A

Proof of Theorem find

Step	Hyp	Ref	Expression
1		find.1	. . 3 |- (A (_ om /\ (/) e. A /\ A.x e. A suc x e. A)
2	1	3simp1i 793	. 2 |- A (_ om
3		ax-1 4	. . . . . . . . 9 |- (suc x e. A -> (x e. om -> suc x e. A))
4	3	r19.20si 1709	. . . . . . . 8 |- (A.x e. A suc x e. A -> A.x e. A (x e. om -> suc x e. A))
5		ralcom3 1780	. . . . . . . 8 |- (A.x e. A (x e. om -> suc x e. A) <-> A.x e. om (x e. A -> suc x e. A))
6	4, 5	sylib 198	. . . . . . 7 |- (A.x e. A suc x e. A -> A.x e. om (x e. A -> suc x e. A))
7	6	anim2i 335	. . . . . 6 |- (((/) e. A /\ A.x e. A suc x e. A) -> ((/) e. A /\ A.x e. om (x e. A -> suc x e. A)))
8	7	anim2i 335	. . . . 5 |- ((A (_ om /\ ((/) e. A /\ A.x e. A suc x e. A)) -> (A (_ om /\ ((/) e. A /\ A.x e. om (x e. A -> suc x e. A))))
9		3anass 781	. . . . 5 |- ((A (_ om /\ (/) e. A /\ A.x e. A suc x e. A) <-> (A (_ om /\ ((/) e. A /\ A.x e. A suc x e. A)))
10		3anass 781	. . . . 5 |- ((A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)) <-> (A (_ om /\ ((/) e. A /\ A.x e. om (x e. A -> suc x e. A))))
11	8, 9, 10	3imtr4 219	. . . 4 |- ((A (_ om /\ (/) e. A /\ A.x e. A suc x e. A) -> (A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)))
12	1, 11	ax-mp 7	. . 3 |- (A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A))
13		peano5 3160	. . . 4 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om (_ A)
14	13	3adant1 799	. . 3 |- ((A (_ om /\ (/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om (_ A)
15	12, 14	ax-mp 7	. 2 |- om (_ A
16	2, 15	eqssi 2082	1 |- A = om

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648   (_ wss 2051  (/)c0 2284  suc csuc 2957  omcom 3138

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786  ax-un 2873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-if 2367  df-pw 2407  df-sn 2417  df-pr 2418  df-tp 2420  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959  df-lim 2960  df-suc 2961  df-om 3139

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