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Theorem ordtr3 4437
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 731 . . . . 5  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  Ord  C )
2 ordelord 4414 . . . . . 6  |-  ( ( Ord  B  /\  A  e.  B )  ->  Ord  A )
32adantlr 695 . . . . 5  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  Ord  A )
4 ordtri1 4425 . . . . 5  |-  ( ( Ord  C  /\  Ord  A )  ->  ( C  C_  A  <->  -.  A  e.  C ) )
51, 3, 4syl2anc 642 . . . 4  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( C  C_  A  <->  -.  A  e.  C ) )
6 ordtr2 4436 . . . . . . 7  |-  ( ( Ord  C  /\  Ord  B )  ->  ( ( C  C_  A  /\  A  e.  B )  ->  C  e.  B ) )
76ancoms 439 . . . . . 6  |-  ( ( Ord  B  /\  Ord  C )  ->  ( ( C  C_  A  /\  A  e.  B )  ->  C  e.  B ) )
87ancomsd 440 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( ( A  e.  B  /\  C  C_  A )  ->  C  e.  B )
)
98expdimp 426 . . . 4  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( C  C_  A  ->  C  e.  B ) )
105, 9sylbird 226 . . 3  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( -.  A  e.  C  ->  C  e.  B ) )
1110orrd 367 . 2  |-  ( ( ( Ord  B  /\  Ord  C )  /\  A  e.  B )  ->  ( A  e.  C  \/  C  e.  B )
)
1211ex 423 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 176    \/ wo 357    /\ wa 358    e. wcel 1684    C_ wss 3152   Ord word 4391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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