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Theorem omsson 7578
 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.
StepHypRef Expression
1 dfom2 7576 . 2 ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
21ssrab3 4043 1 ω ⊆ On
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  {crab 3137   ⊆ wss 3919  Oncon0 6178  Lim wlim 6179  suc csuc 6180  ωcom 7574 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-om 7575 This theorem is referenced by:  limomss  7579  nnon  7580  ordom  7583  omssnlim  7588  omsinds  7594  nnunifi  8766  unblem1  8767  unblem2  8768  unblem3  8769  unblem4  8770  isfinite2  8773  card2inf  9016  ackbij1lem16  9655  ackbij1lem18  9657  fin23lem26  9745  fin23lem27  9748  isf32lem5  9777  fin1a2lem6  9825  pwfseqlem3  10080  tskinf  10189  grothomex  10249  ltsopi  10308  dmaddpi  10310  dmmulpi  10311  2ndcdisj  22064  finminlem  33723
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