Theorem find 4703

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. (Contributed by NM, 22-Feb-2004.) (Proof 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	simp1i 1033	. 2 |- A C_ om
3		3simpc 1023	. . . . 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 5	. . . 4 |- ( (/) e. A /\ A. x e. A suc x e. A )
5		df-ral 2516	. . . . . 6 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
6		alral 2578	. . . . . 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 121	. . . . 5 |- ( A. x e. A suc x e. A -> A. x e. om ( x e. A -> suc x e. A ) )
8	7	anim2i 342	. . . 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 5	. . 3 |- ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) )
10		peano5 4702	. . 3 |- ( ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
11	9, 10	ax-mp 5	. 2 |- om C_ A
12	2, 11	eqssi 3244	1 |- A = om

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   A.wal 1396   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   (/)c0 3496   suc csuc 4468   omcom 4694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-suc 4474  df-iom 4695

This theorem is referenced by: (None)