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Theorem ordtri2or3 4679
Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 4678. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
ordtri2or3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  ( A  i^i  B )  \/  B  =  ( A  i^i  B
) ) )

Proof of Theorem ordtri2or3
StepHypRef Expression
1 ordtri2or2 4678 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  \/  B  C_  A ) )
2 dfss 3335 . . 3  |-  ( A 
C_  B  <->  A  =  ( A  i^i  B ) )
3 dfss5 3546 . . 3  |-  ( B 
C_  A  <->  B  =  ( A  i^i  B ) )
42, 3orbi12i 508 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( A  =  ( A  i^i  B )  \/  B  =  ( A  i^i  B
) ) )
51, 4sylib 189 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  ( A  i^i  B )  \/  B  =  ( A  i^i  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    i^i cin 3319    C_ wss 3320   Ord word 4580
This theorem is referenced by:  ordelinel  4680  mreexexd  13873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584
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