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Theorem omsson 6675
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 6673 . 2  |-  om  =  { x  e.  On  |  suc  x  C_  { y  e.  On  |  -.  Lim  y } }
2 ssrab2 3578 . 2  |-  { x  e.  On  |  suc  x  C_ 
{ y  e.  On  |  -.  Lim  y } }  C_  On
31, 2eqsstri 3527 1  |-  om  C_  On
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   {crab 2811    C_ wss 3469   Oncon0 4871   Lim wlim 4872   suc csuc 4873   omcom 6671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-om 6672
This theorem is referenced by:  limomss  6676  nnon  6677  ordom  6680  omssnlim  6685  nnunifi  7760  unblem1  7761  unblem2  7762  unblem3  7763  unblem4  7764  isfinite2  7767  card2inf  7970  ackbij1lem16  8604  ackbij1lem18  8606  fin23lem26  8694  fin23lem27  8697  isf32lem5  8726  fin1a2lem6  8774  pwfseqlem3  9027  tskinf  9136  grothomex  9196  ltsopi  9255  dmaddpi  9257  dmmulpi  9258  2ndcdisj  19716  omsinds  28862  finminlem  29700
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