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Theorem omssnlim 7340
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 7330 . 2 ω ⊆ On
2 nnlim 7339 . . 3 (𝑥 ∈ ω → ¬ Lim 𝑥)
32rgen 3131 . 2 𝑥 ∈ ω ¬ Lim 𝑥
4 ssrab 3905 . 2 (ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω ¬ Lim 𝑥))
51, 3, 4mpbir2an 704 1 ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wral 3117  {crab 3121  wss 3798  Oncon0 5963  Lim wlim 5964  ωcom 7326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-tr 4976  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-om 7327
This theorem is referenced by: (None)
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