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Theorem sotri3 5073
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 4737 . . . . 5  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
32simprd 449 . . . 4  |-  ( A R B  ->  B  e.  S )
4 soi.1 . . . . . . . 8  |-  R  Or  S
5 sotric 4340 . . . . . . . 8  |-  ( ( R  Or  S  /\  ( C  e.  S  /\  B  e.  S
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
64, 5mpan 651 . . . . . . 7  |-  ( ( C  e.  S  /\  B  e.  S )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
76con2bid 319 . . . . . 6  |-  ( ( C  e.  S  /\  B  e.  S )  ->  ( ( C  =  B  \/  B R C )  <->  -.  C R B ) )
8 breq2 4027 . . . . . . . 8  |-  ( C  =  B  ->  ( A R C  <->  A R B ) )
98biimprd 214 . . . . . . 7  |-  ( C  =  B  ->  ( A R B  ->  A R C ) )
104, 1sotri 5070 . . . . . . . 8  |-  ( ( A R B  /\  B R C )  ->  A R C )
1110expcom 424 . . . . . . 7  |-  ( B R C  ->  ( A R B  ->  A R C ) )
129, 11jaoi 368 . . . . . 6  |-  ( ( C  =  B  \/  B R C )  -> 
( A R B  ->  A R C ) )
137, 12syl6bir 220 . . . . 5  |-  ( ( C  e.  S  /\  B  e.  S )  ->  ( -.  C R B  ->  ( A R B  ->  A R C ) ) )
1413com3r 73 . . . 4  |-  ( A R B  ->  (
( C  e.  S  /\  B  e.  S
)  ->  ( -.  C R B  ->  A R C ) ) )
153, 14mpan2d 655 . . 3  |-  ( A R B  ->  ( C  e.  S  ->  ( -.  C R B  ->  A R C ) ) )
1615com12 27 . 2  |-  ( C  e.  S  ->  ( A R B  ->  ( -.  C R B  ->  A R C ) ) )
17163imp 1145 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023    Or wor 4313    X. cxp 4687
This theorem is referenced by:  archnq  8604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-xp 4695
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