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Theorem sotri2 4906
Description: A transitivity relation. (Read  -. B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1  |-  R  Or  S
soi.2  |-  R  C_  ( S  X.  S
)
Assertion
Ref Expression
sotri2  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  A R C )

Proof of Theorem sotri2
StepHypRef Expression
1 simp2 967 . 2  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  -.  B R A )
2 soi.2 . . . . . . 7  |-  R  C_  ( S  X.  S
)
32brel 4561 . . . . . 6  |-  ( B R C  ->  ( B  e.  S  /\  C  e.  S )
)
433ad2ant3 989 . . . . 5  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  -> 
( B  e.  S  /\  C  e.  S
) )
5 simp1 966 . . . . 5  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  A  e.  S )
6 df-3an 949 . . . . 5  |-  ( ( B  e.  S  /\  C  e.  S  /\  A  e.  S )  <->  ( ( B  e.  S  /\  C  e.  S
)  /\  A  e.  S ) )
74, 5, 6sylanbrc 413 . . . 4  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  -> 
( B  e.  S  /\  C  e.  S  /\  A  e.  S
) )
8 simp3 968 . . . 4  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  B R C )
9 soi.1 . . . . 5  |-  R  Or  S
10 sowlin 4212 . . . . 5  |-  ( ( R  Or  S  /\  ( B  e.  S  /\  C  e.  S  /\  A  e.  S
) )  ->  ( B R C  ->  ( B R A  \/  A R C ) ) )
119, 10mpan 420 . . . 4  |-  ( ( B  e.  S  /\  C  e.  S  /\  A  e.  S )  ->  ( B R C  ->  ( B R A  \/  A R C ) ) )
127, 8, 11sylc 62 . . 3  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  -> 
( B R A  \/  A R C ) )
1312ord 698 . 2  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  -> 
( -.  B R A  ->  A R C ) )
141, 13mpd 13 1  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  A R C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 682    /\ w3a 947    e. wcel 1465    C_ wss 3041   class class class wbr 3899    Or wor 4187    X. cxp 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-iso 4189  df-xp 4515
This theorem is referenced by: (None)
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