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Theorem ordtr3 4586
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 732 . . . . 5  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  Ord  C )
2 ordelord 4563 . . . . . 6  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
32adantlr 696 . . . . 5  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  Ord  A )
4 ordtri1 4574 . . . . 5  |-  ( ( Ord  C  /\  Ord  A )  ->  ( C  C_  A  <->  -.  A  e.  C ) )
51, 3, 4syl2anc 643 . . . 4  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( C  C_  A  <->  -.  A  e.  C ) )
6 ordtr2 4585 . . . . . . 7  |-  ( ( Ord  C  /\  Ord  B )  ->  ( ( C  C_  A  /\  A  e.  B )  ->  C  e.  B ) )
76ancoms 440 . . . . . 6  |-  ( ( Ord  B  /\  Ord  C )  ->  ( ( C  C_  A  /\  A  e.  B )  ->  C  e.  B ) )
87ancomsd 441 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( ( A  e.  B  /\  C  C_  A )  ->  C  e.  B )
)
98expdimp 427 . . . 4  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( C  C_  A  ->  C  e.  B ) )
105, 9sylbird 227 . . 3  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( -.  A  e.  C  ->  C  e.  B ) )
1110orrd 368 . 2  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( A  e.  C  \/  C  e.  B )
)
1211ex 424 1  |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  e.  B  ->  ( A  e.  C  \/  C  e.  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    e. wcel 1721    C_ wss 3280   Ord word 4540
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544
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