Theorem find 4631

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 1008	. 2 |- A C_ om
3		3simpc 998	. . . . 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 2477	. . . . . 6 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
6		alral 2539	. . . . . 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 4630	. . 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 3195	1 |- A = om

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   A.wal 1362   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3153   (/)c0 3446   suc csuc 4396   omcom 4622

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4402  df-iom 4623

This theorem is referenced by: (None)