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Theorem omssnlim 7864
Description: The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of [TakeutiZaring] p. 42. (Contributed by NM, 2-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
omssnlim ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Proof of Theorem omssnlim
StepHypRef Expression
1 omsson 7853 . 2 ω ⊆ On
2 nnlim 7863 . . 3 (𝑥 ∈ ω → ¬ Lim 𝑥)
32rgen 3055 . 2 𝑥 ∈ ω ¬ Lim 𝑥
4 ssrab 4063 . 2 (ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω ¬ Lim 𝑥))
51, 3, 4mpbir2an 708 1 ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wral 3053  {crab 3424  wss 3941  Oncon0 6355  Lim wlim 6356  ωcom 7849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359  df-lim 6360  df-om 7850
This theorem is referenced by: (None)
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