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Theorem omsson 4852
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 4850 . 2  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
2 ssrab2 3430 . 2  |-  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }  C_  On
31, 2eqsstri 3380 1  |-  om  C_  On
Colors of variables: wff set class
Syntax hints:   -. wn 3   {crab 2711    C_ wss 3322   Oncon0 4584   Lim wlim 4585   suc csuc 4586   omcom 4848
This theorem is referenced by:  limomss  4853  nnon  4854  ordom  4857  omssnlim  4862  nnunifi  7361  unblem1  7362  unblem2  7363  unblem3  7364  unblem4  7365  isfinite2  7368  card2inf  7526  ackbij1lem16  8120  ackbij1lem18  8122  fin23lem26  8210  fin23lem27  8213  isf32lem5  8242  fin1a2lem6  8290  pwfseqlem3  8540  tskinf  8649  grothomex  8709  ltsopi  8770  dmaddpi  8772  dmmulpi  8773  2ndcdisj  17524  omsinds  25499  finminlem  26335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849
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