ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sotri2 Unicode version

Theorem sotri2 4750
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 916 . 2  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  -.  B R A )
2 soi.2 . . . . . . 7  |-  R  C_  ( S  X.  S
)
32brel 4420 . . . . . 6  |-  ( B R C  ->  ( B  e.  S  /\  C  e.  S )
)
433ad2ant3 938 . . . . 5  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  -> 
( B  e.  S  /\  C  e.  S
) )
5 simp1 915 . . . . 5  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  A  e.  S )
6 df-3an 898 . . . . 5  |-  ( ( B  e.  S  /\  C  e.  S  /\  A  e.  S )  <->  ( ( B  e.  S  /\  C  e.  S
)  /\  A  e.  S ) )
74, 5, 6sylanbrc 402 . . . 4  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  -> 
( B  e.  S  /\  C  e.  S  /\  A  e.  S
) )
8 simp3 917 . . . 4  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  B R C )
9 soi.1 . . . . 5  |-  R  Or  S
10 sowlin 4085 . . . . 5  |-  ( ( R  Or  S  /\  ( B  e.  S  /\  C  e.  S  /\  A  e.  S
) )  ->  ( B R C  ->  ( B R A  \/  A R C ) ) )
119, 10mpan 408 . . . 4  |-  ( ( B  e.  S  /\  C  e.  S  /\  A  e.  S )  ->  ( B R C  ->  ( B R A  \/  A R C ) ) )
127, 8, 11sylc 60 . . 3  |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  -> 
( B R A  \/  A R C ) )
1312ord 653 . 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 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)
  Copyright terms: Public domain W3C validator