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Theorem peano2b 4264
Description: A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
Assertion
Ref Expression
peano2b (A 𝜔 ↔ suc A 𝜔)

Proof of Theorem peano2b
StepHypRef Expression
1 peano2 4245 . 2 (A 𝜔 → suc A 𝜔)
2 elex 2543 . . . . 5 (suc A 𝜔 → suc A V)
3 sucexb 4173 . . . . 5 (A V ↔ suc A V)
42, 3sylibr 137 . . . 4 (suc A 𝜔 → A V)
5 sucidg 4102 . . . 4 (A V → A suc A)
64, 5syl 14 . . 3 (suc A 𝜔 → A suc A)
7 elnn 4255 . . 3 ((A suc A suc A 𝜔) → A 𝜔)
86, 7mpancom 401 . 2 (suc A 𝜔 → A 𝜔)
91, 8impbii 117 1 (A 𝜔 ↔ suc A 𝜔)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1374  Vcvv 2535  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:  nnmsucr  5982
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