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Theorem sotriOLD 5157
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 4819 . . 3  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
31brel 4819 . . 3  |-  ( B R C  ->  ( B  e.  S  /\  C  e.  S )
)
4 id 19 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
543exp 1150 . . . . 5  |-  ( A  e.  S  ->  ( B  e.  S  ->  ( C  e.  S  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) ) )
65a1dd 42 . . . 4  |-  ( A  e.  S  ->  ( B  e.  S  ->  ( B  e.  S  -> 
( C  e.  S  ->  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) ) ) ) )
76imp43 578 . . 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 463 . 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 4418 . . 3  |-  ( ( R  Or  S  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )  ->  (
( A R B  /\  B R C )  ->  A R C ) )
119, 10mpan 651 . 2  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
128, 11mpcom 32 1  |-  ( ( A R B  /\  B R C )  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1710   _Vcvv 2864    C_ wss 3228   class class class wbr 4104    Or wor 4395    X. cxp 4769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-po 4396  df-so 4397  df-xp 4777
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