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Theorem sotri 5382
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 5037 . . . 4  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
32simpld 457 . . 3  |-  ( A R B  ->  A  e.  S )
41brel 5037 . . 3  |-  ( B R C  ->  ( B  e.  S  /\  C  e.  S )
)
53, 4anim12i 564 . 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 4811 . . . 4  |-  ( ( R  Or  S  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )  ->  (
( A R B  /\  B R C )  ->  A R C ) )
86, 7mpan 668 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
983expb 1195 . 2  |-  ( ( A  e.  S  /\  ( B  e.  S  /\  C  e.  S
) )  ->  (
( A R B  /\  B R C )  ->  A R C ) )
105, 9mpcom 36 1  |-  ( ( A R B  /\  B R C )  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    e. wcel 1823    C_ wss 3461   class class class wbr 4439    Or wor 4788    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-po 4789  df-so 4790  df-xp 4994
This theorem is referenced by:  son2lpi  5383  sotri2  5384  sotri3  5385  ltsonq  9336  ltbtwnnq  9345  nqpr  9381  prlem934  9400  ltexprlem4  9406  reclem2pr  9415  reclem4pr  9417  ltsosr  9460  addgt0sr  9470  supsrlem  9477  axpre-lttrn  9532
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