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Theorem sotr2 4343
Description: A transitivity relation. (Read  B  <_  C and  C  <  D implies  B  <  D.) (Contributed by Mario Carneiro, 10-May-2013.)
Assertion
Ref Expression
sotr2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( -.  C R B  /\  C R D )  ->  B R D ) )

Proof of Theorem sotr2
StepHypRef Expression
1 sotric 4340 . . . . . 6  |-  ( ( R  Or  A  /\  ( C  e.  A  /\  B  e.  A
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
21ancom2s 777 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
323adantr3 1116 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
43con2bid 319 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( C  =  B  \/  B R C )  <->  -.  C R B ) )
5 breq1 4026 . . . . . 6  |-  ( C  =  B  ->  ( C R D  <->  B R D ) )
65biimpd 198 . . . . 5  |-  ( C  =  B  ->  ( C R D  ->  B R D ) )
76a1i 10 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( C  =  B  ->  ( C R D  ->  B R D ) ) )
8 sotr 4336 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )
98exp3a 425 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( B R C  ->  ( C R D  ->  B R D ) ) )
107, 9jaod 369 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( C  =  B  \/  B R C )  ->  ( C R D  ->  B R D ) ) )
114, 10sylbird 226 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( -.  C R B  -> 
( C R D  ->  B R D ) ) )
1211imp3a 420 1  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( -.  C R B  /\  C R D )  ->  B R D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023    Or wor 4313
This theorem is referenced by:  erdszelem8  23729
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-po 4314  df-so 4315
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