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Theorem sotri3 4817
Description: A transitivity relation. (Read A < B and  -. C < B 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
sotri3  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  A R C )

Proof of Theorem sotri3
StepHypRef Expression
1 simp3 945 . 2  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  -.  C R B )
2 soi.2 . . . . . 6  |-  R  C_  ( S  X.  S
)
32brel 4478 . . . . 5  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
433ad2ant2 965 . . . 4  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  -> 
( A  e.  S  /\  B  e.  S
) )
5 simp1 943 . . . 4  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  C  e.  S )
6 df-3an 926 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S ) )
74, 5, 6sylanbrc 408 . . 3  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
8 simp2 944 . . 3  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  A R B )
9 soi.1 . . . 4  |-  R  Or  S
10 sowlin 4138 . . . 4  |-  ( ( R  Or  S  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )  ->  ( A R B  ->  ( A R C  \/  C R B ) ) )
119, 10mpan 415 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A R B  ->  ( A R C  \/  C R B ) ) )
127, 8, 11sylc 61 . 2  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  -> 
( A R C  \/  C R B ) )
131, 12ecased 1285 1  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  A R C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664    /\ w3a 924    e. wcel 1438    C_ wss 2997   class class class wbr 3837    Or wor 4113    X. cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-iso 4115  df-xp 4434
This theorem is referenced by: (None)
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