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Theorem sotri 5261
Description: A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1  |-  R  Or  S
soi.2  |-  R  C_  ( S  X.  S
)
Assertion
Ref Expression
sotri  |-  ( ( A R B  /\  B R C )  ->  A R C )

Proof of Theorem sotri
StepHypRef Expression
1 soi.2 . . . . 5  |-  R  C_  ( S  X.  S
)
21brel 4926 . . . 4  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
32simpld 446 . . 3  |-  ( A R B  ->  A  e.  S )
41brel 4926 . . 3  |-  ( B R C  ->  ( B  e.  S  /\  C  e.  S )
)
53, 4anim12i 550 . 2  |-  ( ( A R B  /\  B R C )  -> 
( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) ) )
6 soi.1 . . . 4  |-  R  Or  S
7 sotr 4525 . . . 4  |-  ( ( R  Or  S  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )  ->  (
( A R B  /\  B R C )  ->  A R C ) )
86, 7mpan 652 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
983expb 1154 . 2  |-  ( ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) )  ->  (
( A R B  /\  B R C )  ->  A R C ) )
105, 9mpcom 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 1725    C_ wss 3320   class class class wbr 4212    Or wor 4502    X. cxp 4876
This theorem is referenced by:  son2lpi  5262  sotri2  5263  sotri3  5264  son2lpiOLD  5267  ltsonq  8846  ltbtwnnq  8855  nqpr  8891  prlem934  8910  ltexprlem4  8916  reclem2pr  8925  reclem4pr  8927  ltsosr  8969  addgt0sr  8979  supsrlem  8986  axpre-lttrn  9041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-po 4503  df-so 4504  df-xp 4884
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