Theorem omsson 7870

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 7867	. 2 ⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
2	1	ssrab3 4062	1 ⊢ ω ⊆ On

Syntax hints:   → wi 4  ∀wal 1538   ⊆ wss 3931  Oncon0 6357  Lim wlim 6358  ωcom 7866

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-ss 3948  df-om 7867

This theorem is referenced by:  limomss  7871  nnon  7872  ordom  7876  omssnlim  7881  omsinds  7887  nnunifi  9304  unblem1  9305  unblem2  9306  unblem3  9307  unblem4  9308  isfinite2  9311  card2inf  9574  ackbij1lem16  10253  ackbij1lem18  10255  fin23lem26  10344  fin23lem27  10347  isf32lem5  10376  fin1a2lem6  10424  pwfseqlem3  10679  tskinf  10788  grothomex  10848  ltsopi  10907  dmaddpi  10909  dmmulpi  10910  2ndcdisj  23399  finminlem  36341  cantnftermord  43311  omabs2  43323