Theorem omssnlim 7870

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 7859	. 2 ⊢ ω ⊆ On
2		nnlim 7869	. . 3 ⊢ (𝑥 ∈ ω → ¬ Lim 𝑥)
3	2	rgen 3064	. 2 ⊢ ∀𝑥 ∈ ω ¬ Lim 𝑥
4		ssrab 4071	. 2 ⊢ (ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥} ↔ (ω ⊆ On ∧ ∀𝑥 ∈ ω ¬ Lim 𝑥))
5	1, 3, 4	mpbir2an 710	1 ⊢ ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Syntax hints:  ¬ wn 3  ∀wral 3062  {crab 3433   ⊆ wss 3949  Oncon0 6365  Lim wlim 6366  ωcom 7855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-om 7856

This theorem is referenced by: (None)