MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omsson Unicode version

Theorem omsson 4816
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 4814 . 2  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
2 ssrab2 3396 . 2  |-  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }  C_  On
31, 2eqsstri 3346 1  |-  om  C_  On
Colors of variables: wff set class
Syntax hints:   -. wn 3   {crab 2678    C_ wss 3288   Oncon0 4549   Lim wlim 4550   suc csuc 4551   omcom 4812
This theorem is referenced by:  limomss  4817  nnon  4818  ordom  4821  omssnlim  4826  nnunifi  7325  unblem1  7326  unblem2  7327  unblem3  7328  unblem4  7329  isfinite2  7332  card2inf  7487  ackbij1lem16  8079  ackbij1lem18  8081  fin23lem26  8169  fin23lem27  8172  isf32lem5  8201  fin1a2lem6  8249  pwfseqlem3  8499  tskinf  8608  grothomex  8668  ltsopi  8729  dmaddpi  8731  dmmulpi  8732  2ndcdisj  17480  omsinds  25441  finminlem  26219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-tr 4271  df-eprel 4462  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813
  Copyright terms: Public domain W3C validator