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Theorem sotr2 4473
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 4470 . . . . . 6  |-  ( ( R  Or  A  /\  ( C  e.  A  /\  B  e.  A
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
21ancom2s 778 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
323adantr3 1118 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
43con2bid 320 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( C  =  B  \/  B R C )  <->  -.  C R B ) )
5 breq1 4156 . . . . . 6  |-  ( C  =  B  ->  ( C R D  <->  B R D ) )
65biimpd 199 . . . . 5  |-  ( C  =  B  ->  ( C R D  ->  B R D ) )
76a1i 11 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( C  =  B  ->  ( C R D  ->  B R D ) ) )
8 sotr 4466 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )
98exp3a 426 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( B R C  ->  ( C R D  ->  B R D ) ) )
107, 9jaod 370 . . 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 227 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( -.  C R B  -> 
( C R D  ->  B R D ) ) )
1211imp3a 421 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4153    Or wor 4443
This theorem is referenced by:  erdszelem8  24663
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
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-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-po 4444  df-so 4445
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