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Theorem sotr2 4838
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 4835 . . . . . 6  |-  ( ( R  Or  A  /\  ( C  e.  A  /\  B  e.  A
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
21ancom2s 802 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
323adantr3 1157 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
43con2bid 329 . . 3  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( C  =  B  \/  B R C )  <->  -.  C R B ) )
5 breq1 4459 . . . . . 6  |-  ( C  =  B  ->  ( C R D  <->  B R D ) )
65biimpd 207 . . . . 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 4831 . . . . 5  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  (
( B R C  /\  C R D )  ->  B R D ) )
98expd 436 . . . 4  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( B R C  ->  ( C R D  ->  B R D ) ) )
107, 9jaod 380 . . 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 235 . 2  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A  /\  D  e.  A
) )  ->  ( -.  C R B  -> 
( C R D  ->  B R D ) ) )
1211impd 431 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   class class class wbr 4456    Or wor 4808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-po 4809  df-so 4810
This theorem is referenced by:  erdszelem8  28926
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