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Theorem omsson 4660
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 4658 . 2  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
2 ssrab2 3258 . 2  |-  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }  C_  On
31, 2eqsstri 3208 1  |-  om  C_  On
Colors of variables: wff set class
Syntax hints:   -. wn 3   {crab 2547    C_ wss 3152   Oncon0 4392   Lim wlim 4393   suc csuc 4394   omcom 4656
This theorem is referenced by:  limomss  4661  nnon  4662  ordom  4665  omssnlim  4670  nnunifi  7108  unblem1  7109  unblem2  7110  unblem3  7111  unblem4  7112  isfinite2  7115  card2inf  7269  ackbij1lem16  7861  ackbij1lem18  7863  fin23lem26  7951  fin23lem27  7954  isf32lem5  7983  fin1a2lem6  8031  pwfseqlem3  8282  tskinf  8391  grothomex  8451  ltsopi  8512  dmaddpi  8514  dmmulpi  8515  2ndcdisj  17182  omsinds  24219  finminlem  26231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657
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