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Theorem ordtri2or2 4505
Description: A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)
Assertion
Ref Expression
ordtri2or2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  \/  B  C_  A ) )

Proof of Theorem ordtri2or2
StepHypRef Expression
1 ordtri2or 4504 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  B  C_  A ) )
2 ordelss 4424 . . . . 5  |-  ( ( Ord  B  /\  A  e.  B )  ->  A  C_  B )
32ex 423 . . . 4  |-  ( Ord 
B  ->  ( A  e.  B  ->  A  C_  B ) )
43orim1d 812 . . 3  |-  ( Ord 
B  ->  ( ( A  e.  B  \/  B  C_  A )  -> 
( A  C_  B  \/  B  C_  A ) ) )
54adantl 452 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  B  C_  A )  -> 
( A  C_  B  \/  B  C_  A ) ) )
61, 5mpd 14 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  \/  B  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    e. wcel 1696    C_ wss 3165   Ord word 4407
This theorem is referenced by:  ordtri2or3  4506  ordssun  4508  ordequn  4509  ordunpr  4633  ackbij2  7885  sornom  7919  fin23lem23  7968  isf32lem2  7996  fpwwe2lem10  8277  hfun  24880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411
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