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Theorem omsson 4763
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 4761 . 2  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
2 ssrab2 3344 . 2  |-  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }  C_  On
31, 2eqsstri 3294 1  |-  om  C_  On
Colors of variables: wff set class
Syntax hints:   -. wn 3   {crab 2632    C_ wss 3238   Oncon0 4495   Lim wlim 4496   suc csuc 4497   omcom 4759
This theorem is referenced by:  limomss  4764  nnon  4765  ordom  4768  omssnlim  4773  nnunifi  7255  unblem1  7256  unblem2  7257  unblem3  7258  unblem4  7259  isfinite2  7262  card2inf  7416  ackbij1lem16  8008  ackbij1lem18  8010  fin23lem26  8098  fin23lem27  8101  isf32lem5  8130  fin1a2lem6  8178  pwfseqlem3  8429  tskinf  8538  grothomex  8598  ltsopi  8659  dmaddpi  8661  dmmulpi  8662  2ndcdisj  17399  omsinds  24960  finminlem  25738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760
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