Theorem find 3788

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. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Hypothesis

Ref	Expression
find.1	|- (A C_ 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 C_ om /\ (/) e. A /\ A.x e. A suc x e. A)
2	1	3simp1i 881	. 2 |- A C_ om
3		3simpc 870	. . . . 5 |- ((A C_ om /\ (/) e. A /\ A.x e. A suc x e. A) -> ((/) e. A /\ A.x e. A suc x e. A))
4	1, 3	ax-mp 7	. . . 4 |- ((/) e. A /\ A.x e. A suc x e. A)
5		df-ral 1943	. . . . . 6 |- (A.x e. A suc x e. A <-> A.x(x e. A -> suc x e. A))
6		alral 1987	. . . . . 6 |- (A.x(x e. A -> suc x e. A) -> A.x e. om (x e. A -> suc x e. A))
7	5, 6	sylbi 215	. . . . 5 |- (A.x e. A suc x e. A -> A.x e. om (x e. A -> suc x e. A))
8	7	anim2i 360	. . . 4 |- (((/) e. A /\ A.x e. A suc x e. A) -> ((/) e. A /\ A.x e. om (x e. A -> suc x e. A)))
9	4, 8	ax-mp 7	. . 3 |- ((/) e. A /\ A.x e. om (x e. A -> suc x e. A))
10		peano5 3786	. . 3 |- (((/) e. A /\ A.x e. om (x e. A -> suc x e. A)) -> om C_ A)
11	9, 10	ax-mp 7	. 2 |- om C_ A
12	2, 11	eqssi 2465	1 |- A = om

Syntax hints:   -> wi 3   /\ wa 239   /\ w3a 855  A.wal 1134   = wceq 1136   e. wcel 1138  A.wral 1939   C_ wss 2426  (/)c0 2701  suc csuc 3474  omcom 3760

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-13 1149  ax-14 1150  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702  ax-sep 3253  ax-nul 3260  ax-pow 3296  ax-pr 3339  ax-un 3601

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-3or 856  df-3an 857  df-ex 1165  df-sb 1374  df-eu 1613  df-mo 1614  df-clab 1709  df-cleq 1714  df-clel 1717  df-ne 1856  df-ral 1943  df-rex 1944  df-rab 1946  df-v 2127  df-dif 2430  df-un 2433  df-in 2436  df-ss 2438  df-pss 2440  df-nul 2702  df-if 2807  df-pw 2859  df-sn 2873  df-pr 2874  df-tp 2876  df-op 2877  df-uni 3000  df-br 3159  df-opab 3214  df-tr 3230  df-eprel 3398  df-po 3406  df-so 3419  df-fr 3440  df-we 3459  df-ord 3475  df-on 3476  df-lim 3477  df-suc 3478  df-om 3761

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