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Theorem sotriOLD 5229
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 4889 . . 3  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
31brel 4889 . . 3  |-  ( B R C  ->  ( B  e.  S  /\  C  e.  S )
)
4 id 20 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
543exp 1152 . . . . 5  |-  ( A  e.  S  ->  ( B  e.  S  ->  ( C  e.  S  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) ) )
65a1dd 44 . . . 4  |-  ( A  e.  S  ->  ( B  e.  S  ->  ( B  e.  S  -> 
( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) ) ) )
76imp43 579 . . 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 464 . 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 4489 . . 3  |-  ( ( R  Or  S  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )  ->  (
( A R B  /\  B R C )  ->  A R C ) )
119, 10mpan 652 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
128, 11mpcom 34 1  |-  ( ( A R B  /\  B R C )  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721   _Vcvv 2920    C_ wss 3284   class class class wbr 4176    Or wor 4466    X. cxp 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-po 4467  df-so 4468  df-xp 4847
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