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Theorem ordtri2 4913
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 4907 . . . . 5  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  C_  A  <->  ( B  e.  A  \/  B  =  A ) ) )
2 eqcom 2476 . . . . . . 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 4911 . . . 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 1379    e. wcel 1767    C_ wss 3476   Ord word 4877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881
This theorem is referenced by:  ord0eln0  4932  oaord  7193  omord2  7213  oeord  7234  nnaord  7265  nnmord  7278
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