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Theorem ordtri2 4559
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 4553 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  ( B  e.  A  \/  B  =  A ) ) )
2 eqcom 2391 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
32orbi2i 506 . . . . . 6  |-  ( ( B  e.  A  \/  B  =  A )  <->  ( B  e.  A  \/  A  =  B )
)
4 orcom 377 . . . . . 6  |-  ( ( B  e.  A  \/  A  =  B )  <->  ( A  =  B  \/  B  e.  A )
)
53, 4bitri 241 . . . . 5  |-  ( ( B  e.  A  \/  B  =  A )  <->  ( A  =  B  \/  B  e.  A )
)
61, 5syl6bb 253 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  ( A  =  B  \/  B  e.  A ) ) )
7 ordtri1 4557 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  -.  A  e.  B ) )
86, 7bitr3d 247 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( ( A  =  B  \/  B  e.  A )  <->  -.  A  e.  B ) )
98ancoms 440 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  =  B  \/  B  e.  A )  <->  -.  A  e.  B ) )
109con2bid 320 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265   Ord word 4523
This theorem is referenced by:  ord0eln0  4578  oaord  6728  omord2  6748  oeord  6769  nnaord  6800  nnmord  6813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-tr 4246  df-eprel 4437  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527
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