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Theorem peano2b 7066
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 limom 7065 . 2 Lim ω
2 limsuc 7034 . 2 (Lim ω → (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω))
31, 2ax-mp 5 1 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1988  Lim wlim 5712  suc csuc 5713  ωcom 7050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-om 7051
This theorem is referenced by:  nnsuc  7067  peano2  7071  peano5  7074  frsuc  7517  frsucmptn  7519  nnaordi  7683  nnmsucr  7690  omsmolem  7718  php  8129  php4  8132  unblem1  8197  isfinite2  8203  inf0  8503  inf3lem1  8510  inf3lem5  8514  cantnfp1lem3  8562  cantnflem1  8571  itunisuc  9226  ituniiun  9229  indpi  9714  rdgeqoa  33189
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