Theorem omsson 7586

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		dfom2 7584	. 2 ⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
2	1	ssrab3 4059	1 ⊢ ω ⊆ On

Syntax hints:  ¬ wn 3  {crab 3144   ⊆ wss 3938  Oncon0 6193  Lim wlim 6194  suc csuc 6195  ωcom 7582

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-om 7583

This theorem is referenced by:  limomss  7587  nnon  7588  ordom  7591  omssnlim  7596  omsinds  7602  nnunifi  8771  unblem1  8772  unblem2  8773  unblem3  8774  unblem4  8775  isfinite2  8778  card2inf  9021  ackbij1lem16  9659  ackbij1lem18  9661  fin23lem26  9749  fin23lem27  9752  isf32lem5  9781  fin1a2lem6  9829  pwfseqlem3  10084  tskinf  10193  grothomex  10253  ltsopi  10312  dmaddpi  10314  dmmulpi  10315  2ndcdisj  22066  finminlem  33668