Theorem omsson 7891

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 7888	. 2 ⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
2	1	ssrab3 4092	1 ⊢ ω ⊆ On

Syntax hints:   → wi 4  ∀wal 1535   ⊆ wss 3963  Oncon0 6386  Lim wlim 6387  ωcom 7887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-ss 3980  df-om 7888

This theorem is referenced by:  limomss  7892  nnon  7893  ordom  7897  omssnlim  7902  omsinds  7908  omsindsOLD  7909  nnunifi  9325  unblem1  9326  unblem2  9327  unblem3  9328  unblem4  9329  isfinite2  9332  card2inf  9593  ackbij1lem16  10272  ackbij1lem18  10274  fin23lem26  10363  fin23lem27  10366  isf32lem5  10395  fin1a2lem6  10443  pwfseqlem3  10698  tskinf  10807  grothomex  10867  ltsopi  10926  dmaddpi  10928  dmmulpi  10929  2ndcdisj  23480  finminlem  36301  cantnftermord  43310  omabs2  43322