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Theorem onsseleq 4871
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 4840 . 2  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4840 . 2  |-  ( B  e.  On  ->  Ord  B )
3 ordsseleq 4859 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
41, 2, 3syl2an 477 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 1370    e. wcel 1758    C_ wss 3439   Ord word 4829   Oncon0 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834
This theorem is referenced by:  onsseli  4944  on0eqel  4947  onmindif2  6536  omword  7122  oeword  7142  oewordi  7143  dffi3  7796  cantnflem1d  8011  cantnflem1  8012  cantnflem1dOLD  8034  cantnflem1OLD  8035  r1ord3g  8101  alephdom  8366  cardaleph  8374  cfsmolem  8554  ttukeylem5  8797  alephreg  8861  inar1  9057  gruina  9100  om2uzlt2i  11895
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