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Theorem sotri3 5233
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 4886 . . . . 5  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
32simprd 465 . . . 4  |-  ( A R B  ->  B  e.  S )
4 soi.1 . . . . . . . 8  |-  R  Or  S
5 sotric 4784 . . . . . . . 8  |-  ( ( R  Or  S  /\  ( C  e.  S  /\  B  e.  S
) )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
64, 5mpan 677 . . . . . . 7  |-  ( ( C  e.  S  /\  B  e.  S )  ->  ( C R B  <->  -.  ( C  =  B  \/  B R C ) ) )
76con2bid 331 . . . . . 6  |-  ( ( C  e.  S  /\  B  e.  S )  ->  ( ( C  =  B  \/  B R C )  <->  -.  C R B ) )
8 breq2 4409 . . . . . . . 8  |-  ( C  =  B  ->  ( A R C  <->  A R B ) )
98biimprd 227 . . . . . . 7  |-  ( C  =  B  ->  ( A R B  ->  A R C ) )
104, 1sotri 5230 . . . . . . . 8  |-  ( ( A R B  /\  B R C )  ->  A R C )
1110expcom 437 . . . . . . 7  |-  ( B R C  ->  ( A R B  ->  A R C ) )
129, 11jaoi 381 . . . . . 6  |-  ( ( C  =  B  \/  B R C )  -> 
( A R B  ->  A R C ) )
137, 12syl6bir 233 . . . . 5  |-  ( ( C  e.  S  /\  B  e.  S )  ->  ( -.  C R B  ->  ( A R B  ->  A R C ) ) )
1413com3r 82 . . . 4  |-  ( A R B  ->  (
( C  e.  S  /\  B  e.  S
)  ->  ( -.  C R B  ->  A R C ) ) )
153, 14mpan2d 681 . . 3  |-  ( A R B  ->  ( C  e.  S  ->  ( -.  C R B  ->  A R C ) ) )
1615com12 32 . 2  |-  ( C  e.  S  ->  ( A R B  ->  ( -.  C R B  ->  A R C ) ) )
17163imp 1203 1  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    C_ wss 3406   class class class wbr 4405    Or wor 4757    X. cxp 4835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-po 4758  df-so 4759  df-xp 4843
This theorem is referenced by:  archnq  9410
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