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Theorem ordtri2or2 4489
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 4488 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  B  C_  A ) )
2 ordelss 4408 . . . . 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 1684    C_ wss 3152   Ord word 4391
This theorem is referenced by:  ordtri2or3  4490  ordssun  4492  ordequn  4493  ordunpr  4617  ackbij2  7869  sornom  7903  fin23lem23  7952  isf32lem2  7980  fpwwe2lem10  8261  hfun  24219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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