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Theorem peano2b 4280
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 4261 . 2 (A 𝜔 → suc A 𝜔)
2 elex 2560 . . . . 5 (suc A 𝜔 → suc A V)
3 sucexb 4189 . . . . 5 (A V ↔ suc A V)
42, 3sylibr 137 . . . 4 (suc A 𝜔 → A V)
5 sucidg 4119 . . . 4 (A V → A suc A)
64, 5syl 14 . . 3 (suc A 𝜔 → A suc A)
7 elnn 4271 . . 3 ((A suc A suc A 𝜔) → A 𝜔)
86, 7mpancom 399 . 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 1390  Vcvv 2551  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:  nnmsucr  6006
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