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Theorem ordtri2or 5537
Description: A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
ordtri2or  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  B  C_  A ) )

Proof of Theorem ordtri2or
StepHypRef Expression
1 ordtri1 5475 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
21ancoms 454 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
32biimprd 226 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  A  e.  B  ->  B 
C_  A ) )
43orrd 379 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  B  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    e. wcel 1870    C_ wss 3442   Ord word 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-ord 5445
This theorem is referenced by:  ordtri2or2  5538  onun2i  5557  ordunisuc2  6685  oaass  7270  alephdom  8510  iscard3  8522  nofulllem5  30380
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