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Theorem sotri2 4904
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 965 . 2  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  -.  B R A )
2 soi.2 . . . . . . 7  |-  R  C_  ( S  X.  S
)
32brel 4559 . . . . . 6  |-  ( B R C  ->  ( B  e.  S  /\  C  e.  S )
)
433ad2ant3 987 . . . . 5  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  -> 
( B  e.  S  /\  C  e.  S
) )
5 simp1 964 . . . . 5  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  A  e.  S )
6 df-3an 947 . . . . 5  |-  ( ( B  e.  S  /\  C  e.  S  /\  A  e.  S )  <->  ( ( B  e.  S  /\  C  e.  S
)  /\  A  e.  S ) )
74, 5, 6sylanbrc 411 . . . 4  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  -> 
( B  e.  S  /\  C  e.  S  /\  A  e.  S
) )
8 simp3 966 . . . 4  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  B R C )
9 soi.1 . . . . 5  |-  R  Or  S
10 sowlin 4210 . . . . 5  |-  ( ( R  Or  S  /\  ( B  e.  S  /\  C  e.  S  /\  A  e.  S
) )  ->  ( B R C  ->  ( B R A  \/  A R C ) ) )
119, 10mpan 418 . . . 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 696 . 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 680    /\ w3a 945    e. wcel 1463    C_ wss 3039   class class class wbr 3897    Or wor 4185    X. cxp 4505
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 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-iso 4187  df-xp 4513
This theorem is referenced by: (None)
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