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Theorem peano2b 7455
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 7454 . 2 Lim ω
2 limsuc 7423 . 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 207  wcel 2080  Lim wlim 6070  suc csuc 6071  ωcom 7439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pr 5224  ax-un 7322
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-tr 5067  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-om 7440
This theorem is referenced by:  nnsuc  7456  peano2  7461  peano5  7464  frsuc  7927  frsucmptn  7929  nnaordi  8097  nnmsucr  8104  omsmolem  8133  php  8551  php4  8554  unblem1  8619  isfinite2  8625  inf0  8933  inf3lem1  8940  inf3lem5  8944  cantnfp1lem3  8992  cantnflem1  9001  itunisuc  9690  ituniiun  9693  indpi  10178  rdgeqoa  34195
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