Theorem find 4691

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 1030	. 2 |- A C_ om
3		3simpc 1020	. . . . 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 2513	. . . . . 6 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
6		alral 2575	. . . . . 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 4690	. . 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 3240	1 |- A = om

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   A.wal 1393   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   (/)c0 3491   suc csuc 4456   omcom 4682

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3889  df-int 3924  df-suc 4462  df-iom 4683

This theorem is referenced by: (None)