Theorem omssnlim 7832

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 7821	. 2 ⊢ ω ⊆ On
2		nnlim 7831	. . 3 ⊢ (𝑥 ∈ ω → ¬ Lim 𝑥)
3	2	rgen 3053	. 2 ⊢ ∀𝑥 ∈ ω ¬ Lim 𝑥
4		ssrab 4011	. 2 ⊢ (ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω ¬ Lim 𝑥))
5	1, 3, 4	mpbir2an 712	1 ⊢ ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Syntax hints:  ¬ wn 3  ∀wral 3051  {crab 3389   ⊆ wss 3889  Oncon0 6323  Lim wlim 6324  ωcom 7817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-lim 6328  df-om 7818

This theorem is referenced by: (None)