Theorem omssnlim 7821

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 7810	. 2 ⊢ ω ⊆ On
2		nnlim 7820	. . 3 ⊢ (𝑥 ∈ ω → ¬ Lim 𝑥)
3	2	rgen 3055	. 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 717	1 ⊢ ω ⊆ {𝑥 ∈ On ∣ ¬ Lim 𝑥}

Syntax hints:  ¬ wn 3  ∀wral 3053  {crab 3391   ⊆ wss 3883  Oncon0 6310  Lim wlim 6311  ωcom 7806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314  df-lim 6315  df-om 7807

This theorem is referenced by: (None)