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Theorem omsson 6677
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 6675 . 2  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
2 ssrab2 3571 . 2  |-  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }  C_  On
31, 2eqsstri 3519 1  |-  om  C_  On
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   {crab 2808    C_ wss 3461   Oncon0 4867   Lim wlim 4868   suc csuc 4869   omcom 6673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-om 6674
This theorem is referenced by:  limomss  6678  nnon  6679  ordom  6682  omssnlim  6687  nnunifi  7763  unblem1  7764  unblem2  7765  unblem3  7766  unblem4  7767  isfinite2  7770  card2inf  7973  ackbij1lem16  8606  ackbij1lem18  8608  fin23lem26  8696  fin23lem27  8699  isf32lem5  8728  fin1a2lem6  8776  pwfseqlem3  9027  tskinf  9136  grothomex  9196  ltsopi  9255  dmaddpi  9257  dmmulpi  9258  2ndcdisj  20123  omsinds  29539  finminlem  30376
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