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Theorem ordtri2or 4819
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 4757 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
21ancoms 453 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
32biimprd 223 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  A  e.  B  ->  B 
C_  A ) )
43orrd 378 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  B  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    e. wcel 1756    C_ wss 3333   Ord word 4723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-tr 4391  df-eprel 4637  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727
This theorem is referenced by:  ordtri2or2  4820  onun2i  4839  ordunisuc2  6460  oaass  7005  alephdom  8256  iscard3  8268  nofulllem5  27852
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