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Theorem ordtr3 5487
Description: Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.)
Assertion
Ref Expression
ordtr3  |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  e.  B  ->  ( A  e.  C  \/  C  e.  B ) ) )

Proof of Theorem ordtr3
StepHypRef Expression
1 simplr 760 . . . . 5  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  Ord  C )
2 ordelord 5464 . . . . . 6  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
32adantlr 719 . . . . 5  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  Ord  A )
4 ordtri1 5475 . . . . 5  |-  ( ( Ord  C  /\  Ord  A )  ->  ( C  C_  A  <->  -.  A  e.  C ) )
51, 3, 4syl2anc 665 . . . 4  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( C  C_  A  <->  -.  A  e.  C ) )
6 ordtr2 5486 . . . . . . 7  |-  ( ( Ord  C  /\  Ord  B )  ->  ( ( C  C_  A  /\  A  e.  B )  ->  C  e.  B ) )
76ancoms 454 . . . . . 6  |-  ( ( Ord  B  /\  Ord  C )  ->  ( ( C  C_  A  /\  A  e.  B )  ->  C  e.  B ) )
87ancomsd 455 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( ( A  e.  B  /\  C  C_  A )  ->  C  e.  B )
)
98expdimp 438 . . . 4  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( C  C_  A  ->  C  e.  B ) )
105, 9sylbird 238 . . 3  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( -.  A  e.  C  ->  C  e.  B ) )
1110orrd 379 . 2  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( A  e.  C  \/  C  e.  B )
)
1211ex 435 1  |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  e.  B  ->  ( A  e.  C  \/  C  e.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    e. wcel 1870    C_ wss 3442   Ord word 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-ord 5445
This theorem is referenced by: (None)
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