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Theorem find 4249
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 ⊆ 𝜔 A x A suc x A)
Assertion
Ref Expression
find A = 𝜔
Distinct variable group:   x,A

Proof of Theorem find
StepHypRef Expression
1 find.1 . . 3 (A ⊆ 𝜔 A x A suc x A)
21simp1i 901 . 2 A ⊆ 𝜔
3 3simpc 891 . . . . 5 ((A ⊆ 𝜔 A x A suc x A) → (∅ A x A suc x A))
41, 3ax-mp 7 . . . 4 (∅ A x A suc x A)
5 df-ral 2289 . . . . . 6 (x A suc x Ax(x A → suc x A))
6 alral 2345 . . . . . 6 (x(x A → suc x A) → x 𝜔 (x A → suc x A))
75, 6sylbi 114 . . . . 5 (x A suc x Ax 𝜔 (x A → suc x A))
87anim2i 324 . . . 4 ((∅ A x A suc x A) → (∅ A x 𝜔 (x A → suc x A)))
94, 8ax-mp 7 . . 3 (∅ A x 𝜔 (x A → suc x A))
10 peano5 4248 . . 3 ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
119, 10ax-mp 7 . 2 𝜔 ⊆ A
122, 11eqssi 2938 1 A = 𝜔
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873  wal 1226   = wceq 1228   wcel 1374  wral 2284  wss 2894  c0 3201  suc csuc 4051  𝜔com 4240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241
This theorem is referenced by: (None)
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