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Theorem onsseleq 4747
Description: Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
onsseleq  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )

Proof of Theorem onsseleq
StepHypRef Expression
1 eloni 4716 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4716 . 2  |-  ( B  e.  On  ->  Ord  B )
3 ordsseleq 4735 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
41, 2, 3syl2an 474 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1362    e. wcel 1755    C_ wss 3316   Ord word 4705   Oncon0 4706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-tr 4374  df-eprel 4619  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710
This theorem is referenced by:  onsseli  4820  on0eqel  4823  onmindif2  6412  omword  6997  oeword  7017  oewordi  7018  dffi3  7669  cantnflem1d  7884  cantnflem1  7885  cantnflem1dOLD  7907  cantnflem1OLD  7908  r1ord3g  7974  alephdom  8239  cardaleph  8247  cfsmolem  8427  ttukeylem5  8670  alephreg  8734  inar1  8930  gruina  8973  om2uzlt2i  11758
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