Theorem omsson 7812

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 7809	. 2 ⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
2	1	ssrab3 4034	1 ⊢ ω ⊆ On

Syntax hints:   → wi 4  ∀wal 1539   ⊆ wss 3901  Oncon0 6317  Lim wlim 6318  ωcom 7808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-ss 3918  df-om 7809

This theorem is referenced by:  limomss  7813  nnon  7814  ordom  7818  omssnlim  7823  omsinds  7829  nnunifi  9191  unblem1  9192  unblem2  9193  unblem3  9194  unblem4  9195  isfinite2  9198  card2inf  9460  ackbij1lem16  10144  ackbij1lem18  10146  fin23lem26  10235  fin23lem27  10238  isf32lem5  10267  fin1a2lem6  10315  pwfseqlem3  10571  tskinf  10680  grothomex  10740  ltsopi  10799  dmaddpi  10801  dmmulpi  10802  2ndcdisj  23400  finminlem  36512  cantnftermord  43558  omabs2  43570