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Theorem sotri3 5266
Description: A transitivity relation. (Read  A  <  B and  B  <_  C implies  A  <  C.) (Contributed by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1  |-  R  Or  S
soi.2  |-  R  C_  ( S  X.  S
)
Assertion
Ref Expression
sotri3  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  A R C )

Proof of Theorem sotri3
StepHypRef Expression
1 soi.2 . . . . . 6  |-  R  C_  ( S  X.  S
)
21brel 4928 . . . . 5  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
32simprd 451 . . . 4  |-  ( A R B  ->  B  e.  S )
4 soi.1 . . . . . . . 8  |-  R  Or  S
5 sotric 4531 . . . . . . . 8  |-  ( ( R  Or  S  /\  ( C  e.  S  /\  B  e.  S
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
64, 5mpan 653 . . . . . . 7  |-  ( ( C  e.  S  /\  B  e.  S )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
76con2bid 321 . . . . . 6  |-  ( ( C  e.  S  /\  B  e.  S )  ->  ( ( C  =  B  \/  B R C )  <->  -.  C R B ) )
8 breq2 4218 . . . . . . . 8  |-  ( C  =  B  ->  ( A R C  <->  A R B ) )
98biimprd 216 . . . . . . 7  |-  ( C  =  B  ->  ( A R B  ->  A R C ) )
104, 1sotri 5263 . . . . . . . 8  |-  ( ( A R B  /\  B R C )  ->  A R C )
1110expcom 426 . . . . . . 7  |-  ( B R C  ->  ( A R B  ->  A R C ) )
129, 11jaoi 370 . . . . . 6  |-  ( ( C  =  B  \/  B R C )  -> 
( A R B  ->  A R C ) )
137, 12syl6bir 222 . . . . 5  |-  ( ( C  e.  S  /\  B  e.  S )  ->  ( -.  C R B  ->  ( A R B  ->  A R C ) ) )
1413com3r 76 . . . 4  |-  ( A R B  ->  (
( C  e.  S  /\  B  e.  S
)  ->  ( -.  C R B  ->  A R C ) ) )
153, 14mpan2d 657 . . 3  |-  ( A R B  ->  ( C  e.  S  ->  ( -.  C R B  ->  A R C ) ) )
1615com12 30 . 2  |-  ( C  e.  S  ->  ( A R B  ->  ( -.  C R B  ->  A R C ) ) )
17163imp 1148 1  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  A R C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   class class class wbr 4214    Or wor 4504    X. cxp 4878
This theorem is referenced by:  archnq  8859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-po 4505  df-so 4506  df-xp 4886
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