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Theorem sotr2 4524
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 4521 . . . . . 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 4207 . . . . . 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 4517 . . . . 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 1652    e. wcel 1725   class class class wbr 4204    Or wor 4494
This theorem is referenced by:  erdszelem8  24876
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-po 4495  df-so 4496
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