Theorem omsson 6961

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 6959	. 2 ⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
2		ssrab2 3650	. 2 ⊢ {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}} ⊆ On
3	1, 2	eqsstri 3598	1 ⊢ ω ⊆ On

Syntax hints:  ¬ wn 3  {crab 2900   ⊆ wss 3540  Oncon0 5640  Lim wlim 5641  suc csuc 5642  ωcom 6957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958

This theorem is referenced by:  limomss  6962  nnon  6963  ordom  6966  omssnlim  6971  omsinds  6976  nnunifi  8096  unblem1  8097  unblem2  8098  unblem3  8099  unblem4  8100  isfinite2  8103  card2inf  8343  ackbij1lem16  8940  ackbij1lem18  8942  fin23lem26  9030  fin23lem27  9033  isf32lem5  9062  fin1a2lem6  9110  pwfseqlem3  9361  tskinf  9470  grothomex  9530  ltsopi  9589  dmaddpi  9591  dmmulpi  9592  2ndcdisj  21069  finminlem  31482