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Theorem ordtri2 4766
Description: A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)
Assertion
Ref Expression
ordtri2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )

Proof of Theorem ordtri2
StepHypRef Expression
1 ordsseleq 4760 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  ( B  e.  A  \/  B  =  A ) ) )
2 eqcom 2445 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
32orbi2i 519 . . . . . 6  |-  ( ( B  e.  A  \/  B  =  A )  <->  ( B  e.  A  \/  A  =  B )
)
4 orcom 387 . . . . . 6  |-  ( ( B  e.  A  \/  A  =  B )  <->  ( A  =  B  \/  B  e.  A )
)
53, 4bitri 249 . . . . 5  |-  ( ( B  e.  A  \/  B  =  A )  <->  ( A  =  B  \/  B  e.  A )
)
61, 5syl6bb 261 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  ( A  =  B  \/  B  e.  A ) ) )
7 ordtri1 4764 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
86, 7bitr3d 255 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( ( A  =  B  \/  B  e.  A )  <->  -.  A  e.  B ) )
98ancoms 453 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  =  B  \/  B  e.  A )  <->  -.  A  e.  B ) )
109con2bid 329 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3340   Ord word 4730
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 4425  ax-nul 4433  ax-pr 4543
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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-tr 4398  df-eprel 4644  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734
This theorem is referenced by:  ord0eln0  4785  oaord  6998  omord2  7018  oeord  7039  nnaord  7070  nnmord  7083
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