Theorem omsson 7863

Description: Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
omsson	⊢ ω ⊆ On

Proof of Theorem omsson

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-om 7860	. 2 ⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
2	1	ssrab3 4081	1 ⊢ ω ⊆ On

Syntax hints:   → wi 4  ∀wal 1537   ⊆ wss 3949  Oncon0 6365  Lim wlim 6366  ωcom 7859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-in 3956  df-ss 3966  df-om 7860

This theorem is referenced by:  limomss  7864  nnon  7865  ordom  7869  omssnlim  7874  omsinds  7880  omsindsOLD  7881  nnunifi  9298  unblem1  9299  unblem2  9300  unblem3  9301  unblem4  9302  isfinite2  9305  card2inf  9554  ackbij1lem16  10234  ackbij1lem18  10236  fin23lem26  10324  fin23lem27  10327  isf32lem5  10356  fin1a2lem6  10404  pwfseqlem3  10659  tskinf  10768  grothomex  10828  ltsopi  10887  dmaddpi  10889  dmmulpi  10890  2ndcdisj  23182  finminlem  35508  cantnftermord  42374  omabs2  42386