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Theorem ordtri2 5445
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 5439 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  ( B  e.  A  \/  B  =  A ) ) )
2 eqcom 2411 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
32orbi2i 517 . . . . . 6  |-  ( ( B  e.  A  \/  B  =  A )  <->  ( B  e.  A  \/  A  =  B )
)
4 orcom 385 . . . . . 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 5443 . . . 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 451 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  =  B  \/  B  e.  A )  <->  -.  A  e.  B ) )
109con2bid 327 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 366    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3414   Ord word 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-tr 4490  df-eprel 4734  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 5413
This theorem is referenced by:  ord0eln0  5464  oaord  7233  omord2  7253  oeord  7274  nnaord  7305  nnmord  7318
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