Theorem omsson 7069

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 7067	. 2 ⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
2		ssrab2 3687	. 2 ⊢ {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}} ⊆ On
3	1, 2	eqsstri 3635	1 ⊢ ω ⊆ On

Syntax hints:  ¬ wn 3  {crab 2916   ⊆ wss 3574  Oncon0 5723  Lim wlim 5724  suc csuc 5725  ωcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066

This theorem is referenced by:  limomss  7070  nnon  7071  ordom  7074  omssnlim  7079  omsinds  7084  nnunifi  8211  unblem1  8212  unblem2  8213  unblem3  8214  unblem4  8215  isfinite2  8218  card2inf  8460  ackbij1lem16  9057  ackbij1lem18  9059  fin23lem26  9147  fin23lem27  9150  isf32lem5  9179  fin1a2lem6  9227  pwfseqlem3  9482  tskinf  9591  grothomex  9651  ltsopi  9710  dmaddpi  9712  dmmulpi  9713  2ndcdisj  21259  finminlem  32312