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Theorem sotr2 4359
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 4356 . . . . . 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 4042 . . . . . 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 4352 . . . . 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 1632    e. wcel 1696   class class class wbr 4039    Or wor 4329
This theorem is referenced by:  erdszelem8  23744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-po 4330  df-so 4331
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