Theorem omsson 7710

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 7707	. 2 ⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
2	1	ssrab3 4020	1 ⊢ ω ⊆ On

Syntax hints:   → wi 4  ∀wal 1540   ⊆ wss 3892  Oncon0 6265  Lim wlim 6266  ωcom 7706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-in 3899  df-ss 3909  df-om 7707

This theorem is referenced by:  limomss  7711  nnon  7712  ordom  7716  omssnlim  7721  omsinds  7727  omsindsOLD  7728  nnunifi  9043  unblem1  9044  unblem2  9045  unblem3  9046  unblem4  9047  isfinite2  9050  card2inf  9292  ackbij1lem16  9992  ackbij1lem18  9994  fin23lem26  10082  fin23lem27  10085  isf32lem5  10114  fin1a2lem6  10162  pwfseqlem3  10417  tskinf  10526  grothomex  10586  ltsopi  10645  dmaddpi  10647  dmmulpi  10648  2ndcdisj  22605  finminlem  34503