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Theorem ordtr3 4618
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 4595 . . . . . 6  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
32adantlr 696 . . . . 5  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  Ord  A )
4 ordtri1 4606 . . . . 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 4617 . . . . . . 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 1725    C_ wss 3312   Ord word 4572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576
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