Theorem omsson 7879

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 7876	. 2 ⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
2	1	ssrab3 4078	1 ⊢ ω ⊆ On

Syntax hints:   → wi 4  ∀wal 1531   ⊆ wss 3946  Oncon0 6375  Lim wlim 6376  ωcom 7875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-ss 3963  df-om 7876

This theorem is referenced by:  limomss  7880  nnon  7881  ordom  7885  omssnlim  7890  omsinds  7896  omsindsOLD  7897  nnunifi  9331  unblem1  9332  unblem2  9333  unblem3  9334  unblem4  9335  isfinite2  9338  card2inf  9594  ackbij1lem16  10274  ackbij1lem18  10276  fin23lem26  10364  fin23lem27  10367  isf32lem5  10396  fin1a2lem6  10444  pwfseqlem3  10699  tskinf  10808  grothomex  10868  ltsopi  10927  dmaddpi  10929  dmmulpi  10930  2ndcdisj  23443  finminlem  35978  cantnftermord  42923  omabs2  42935