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Theorem omsson 7111
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 7109 . 2 ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
2 ssrab2 3720 . 2 {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}} ⊆ On
31, 2eqsstri 3668 1 ω ⊆ On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  {crab 2945  wss 3607  Oncon0 5761  Lim wlim 5762  suc csuc 5763  ωcom 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-om 7108
This theorem is referenced by:  limomss  7112  nnon  7113  ordom  7116  omssnlim  7121  omsinds  7126  nnunifi  8252  unblem1  8253  unblem2  8254  unblem3  8255  unblem4  8256  isfinite2  8259  card2inf  8501  ackbij1lem16  9095  ackbij1lem18  9097  fin23lem26  9185  fin23lem27  9188  isf32lem5  9217  fin1a2lem6  9265  pwfseqlem3  9520  tskinf  9629  grothomex  9689  ltsopi  9748  dmaddpi  9750  dmmulpi  9751  2ndcdisj  21307  finminlem  32437
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