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Theorem sotriOLD 5269
Description: A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
soiOLD.1  |-  A  e. 
_V
soiOLD.2  |-  R  Or  S
soiOLD.3  |-  R  C_  ( S  X.  S
)
sotriOLD.4  |-  B  e. 
_V
sotriOLD.5  |-  C  e. 
_V
Assertion
Ref Expression
sotriOLD  |-  ( ( A R B  /\  B R C )  ->  A R C )

Proof of Theorem sotriOLD
StepHypRef Expression
1 soiOLD.3 . . . 4  |-  R  C_  ( S  X.  S
)
21brel 4929 . . 3  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
31brel 4929 . . 3  |-  ( B R C  ->  ( B  e.  S  /\  C  e.  S )
)
4 id 21 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
543exp 1153 . . . . 5  |-  ( A  e.  S  ->  ( B  e.  S  ->  ( C  e.  S  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) ) )
65a1dd 45 . . . 4  |-  ( A  e.  S  ->  ( B  e.  S  ->  ( B  e.  S  -> 
( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) ) ) )
76imp43 580 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( B  e.  S  /\  C  e.  S ) )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
82, 3, 7syl2an 465 . 2  |-  ( ( A R B  /\  B R C )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
9 soiOLD.2 . . 3  |-  R  Or  S
10 sotr 4528 . . 3  |-  ( ( R  Or  S  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )  ->  (
( A R B  /\  B R C )  ->  A R C ) )
119, 10mpan 653 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
128, 11mpcom 35 1  |-  ( ( A R B  /\  B R C )  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726   _Vcvv 2958    C_ wss 3322   class class class wbr 4215    Or wor 4505    X. cxp 4879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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 4216  df-opab 4270  df-po 4506  df-so 4507  df-xp 4887
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