Theorem omssnlim 7702

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

Step	Hyp	Ref	Expression
1		omsson 7691	. 2 ⊢ ω ⊆ On
2		nnlim 7701	. . 3 ⊢ (𝑥 ∈ ω → ¬ Lim 𝑥)
3	2	rgen 3073	. 2 ⊢ ∀𝑥 ∈ ω ¬ Lim 𝑥
4		ssrab 4002	. 2 ⊢ (ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω ¬ Lim 𝑥))
5	1, 3, 4	mpbir2an 707	1 ⊢ ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Syntax hints:  ¬ wn 3  ∀wral 3063  {crab 3067   ⊆ wss 3883  Oncon0 6251  Lim wlim 6252  ωcom 7687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-lim 6256  df-om 7688

This theorem is referenced by: (None)