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Theorem omsson 6480
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  |-  om  C_  On

Proof of Theorem omsson
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfom2 6478 . 2  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
2 ssrab2 3437 . 2  |-  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }  C_  On
31, 2eqsstri 3386 1  |-  om  C_  On
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   {crab 2719    C_ wss 3328   Oncon0 4719   Lim wlim 4720   suc csuc 4721   omcom 6476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-om 6477
This theorem is referenced by:  limomss  6481  nnon  6482  ordom  6485  omssnlim  6490  nnunifi  7563  unblem1  7564  unblem2  7565  unblem3  7566  unblem4  7567  isfinite2  7570  card2inf  7770  ackbij1lem16  8404  ackbij1lem18  8406  fin23lem26  8494  fin23lem27  8497  isf32lem5  8526  fin1a2lem6  8574  pwfseqlem3  8827  tskinf  8936  grothomex  8996  ltsopi  9057  dmaddpi  9059  dmmulpi  9060  2ndcdisj  19060  omsinds  27680  finminlem  28513
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