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Theorem omsson 7237
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 7235 . 2 ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
2 ssrab2 3843 . 2 {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}} ⊆ On
31, 2eqsstri 3791 1 ω ⊆ On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  {crab 3068  wss 3729  Oncon0 5877  Lim wlim 5878  suc csuc 5879  ωcom 7233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-tr 4900  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-om 7234
This theorem is referenced by:  limomss  7238  nnon  7239  ordom  7242  omssnlim  7247  omsinds  7252  nnunifi  8388  unblem1  8389  unblem2  8390  unblem3  8391  unblem4  8392  isfinite2  8395  card2inf  8637  ackbij1lem16  9280  ackbij1lem18  9282  fin23lem26  9370  fin23lem27  9373  isf32lem5  9402  fin1a2lem6  9450  pwfseqlem3  9705  tskinf  9814  grothomex  9874  ltsopi  9933  dmaddpi  9935  dmmulpi  9936  2ndcdisj  21500  finminlem  32666
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