MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  peano2b Structured version   Visualization version   GIF version

Theorem peano2b 7227
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 7226 . 2 Lim ω
2 limsuc 7195 . 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 2144  Lim wlim 5867  suc csuc 5868  ωcom 7211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-tr 4885  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-om 7212
This theorem is referenced by:  nnsuc  7228  peano2  7232  peano5  7235  frsuc  7684  frsucmptn  7686  nnaordi  7851  nnmsucr  7858  omsmolem  7886  php  8299  php4  8302  unblem1  8367  isfinite2  8373  inf0  8681  inf3lem1  8688  inf3lem5  8692  cantnfp1lem3  8740  cantnflem1  8749  itunisuc  9442  ituniiun  9445  indpi  9930  rdgeqoa  33548
  Copyright terms: Public domain W3C validator