Theorem find 6725

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 is a set of natural numbers, zero belongs to , and given any member of the member's successor also belongs to . The conclusion is that every natural number is in . (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypothesis

Ref	Expression
find.1

Assertion

Ref	Expression
find

Distinct variable group:   ,

Proof of Theorem find

Step	Hyp	Ref	Expression
1		find.1	. . 3
2	1	simp1i 1005	. 2
3		3simpc 995	. . . . 5
4	1, 3	ax-mp 5	. . . 4
5		df-ral 2812	. . . . . 6
6		alral 2822	. . . . . 6
7	5, 6	sylbi 195	. . . . 5
8	7	anim2i 569	. . . 4
9	4, 8	ax-mp 5	. . 3
10		peano5 6723	. . 3
11	9, 10	ax-mp 5	. 2
12	2, 11	eqssi 3519	1

Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  A.wal 1393  =wceq 1395  e.wcel 1818  A.wral 2807  C_wss 3475  c0 3784  succsuc 4885  com 6700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-om 6701