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Theorem find 4265
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 912 . 2 A ⊆ 𝜔
3 3simpc 902 . . . . 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 2305 . . . . . 6 (x A suc x Ax(x A → suc x A))
6 alral 2361 . . . . . 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 4264 . . 3 ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
119, 10ax-mp 7 . 2 𝜔 ⊆ A
122, 11eqssi 2955 1 A = 𝜔
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884  wal 1240   = wceq 1242   wcel 1390  wral 2300  wss 2911  c0 3218  suc csuc 4068  𝜔com 4256
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257
This theorem is referenced by: (None)
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