Theorem find 4583

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 1001	. 2 |- A C_ om
3		3simpc 991	. . . . 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 2453	. . . . . 6 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
6		alral 2515	. . . . . 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 120	. . . . 5 |- ( A. x e. A suc x e. A -> A. x e. om ( x e. A -> suc x e. A ) )
8	7	anim2i 340	. . . 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 4582	. . 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 3163	1 |- A = om

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   (/)c0 3414   suc csuc 4350   omcom 4574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575

This theorem is referenced by: (None)