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Theorem sotri3 4751
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 917 . 2  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  -.  C R B )
2 soi.2 . . . . . 6  |-  R  C_  ( S  X.  S
)
32brel 4420 . . . . 5  |-  ( A R B  ->  ( A  e.  S  /\  B  e.  S )
)
433ad2ant2 937 . . . 4  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  -> 
( A  e.  S  /\  B  e.  S
) )
5 simp1 915 . . . 4  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  C  e.  S )
6 df-3an 898 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  <->  ( ( A  e.  S  /\  B  e.  S
)  /\  C  e.  S ) )
74, 5, 6sylanbrc 402 . . 3  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  -> 
( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )
8 simp2 916 . . 3  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  A R B )
9 soi.1 . . . 4  |-  R  Or  S
10 sowlin 4085 . . . 4  |-  ( ( R  Or  S  /\  ( A  e.  S  /\  B  e.  S  /\  C  e.  S
) )  ->  ( A R B  ->  ( A R C  \/  C R B ) ) )
119, 10mpan 408 . . 3  |-  ( ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  ( A R B  ->  ( A R C  \/  C R B ) ) )
127, 8, 11sylc 60 . 2  |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  -> 
( A R C  \/  C R B ) )
131, 12ecased 1255 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 101    \/ wo 639    /\ w3a 896    e. wcel 1409    C_ wss 2945   class class class wbr 3792    Or wor 4060    X. cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-iso 4062  df-xp 4379
This theorem is referenced by: (None)
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