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Theorem ordtr3 4763
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 754 . . . . 5  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  Ord  C )
2 ordelord 4740 . . . . . 6  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
32adantlr 714 . . . . 5  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  Ord  A )
4 ordtri1 4751 . . . . 5  |-  ( ( Ord  C  /\  Ord  A )  ->  ( C  C_  A  <->  -.  A  e.  C ) )
51, 3, 4syl2anc 661 . . . 4  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( C  C_  A  <->  -.  A  e.  C ) )
6 ordtr2 4762 . . . . . . 7  |-  ( ( Ord  C  /\  Ord  B )  ->  ( ( C  C_  A  /\  A  e.  B )  ->  C  e.  B ) )
76ancoms 453 . . . . . 6  |-  ( ( Ord  B  /\  Ord  C )  ->  ( ( C  C_  A  /\  A  e.  B )  ->  C  e.  B ) )
87ancomsd 454 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( ( A  e.  B  /\  C  C_  A )  ->  C  e.  B )
)
98expdimp 437 . . . 4  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( C  C_  A  ->  C  e.  B ) )
105, 9sylbird 235 . . 3  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( -.  A  e.  C  ->  C  e.  B ) )
1110orrd 378 . 2  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( A  e.  C  \/  C  e.  B )
)
1211ex 434 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 184    \/ wo 368    /\ wa 369    e. wcel 1756    C_ wss 3327   Ord word 4717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-tr 4385  df-eprel 4631  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721
This theorem is referenced by: (None)
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