Theorem find 4741

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 1037	. 2 |- A C_ om
3		3simpc 1027	. . . . 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 2533	. . . . . 6 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
6		alral 2595	. . . . . 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 4740	. . 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 3264	1 |- A = om

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   A.wal 1400   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   (/)c0 3520   suc csuc 4505   omcom 4732

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733

This theorem is referenced by: (None)