MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sotri2 Structured version   Unicode version

Theorem sotri2 5255
Description: A transitivity relation. (Read  A  <_  B 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 soi.2 . . . . . 6  |-  R  C_  ( S  X.  S
)
21brel 4918 . . . . 5  |-  ( B R C  ->  ( B  e.  S  /\  C  e.  S )
)
32simpld 446 . . . 4  |-  ( B R C  ->  B  e.  S )
4 soi.1 . . . . . . . 8  |-  R  Or  S
5 sotric 4521 . . . . . . . 8  |-  ( ( R  Or  S  /\  ( B  e.  S  /\  A  e.  S
) )  ->  ( B R A  <->  -.  ( B  =  A  \/  A R B ) ) )
64, 5mpan 652 . . . . . . 7  |-  ( ( B  e.  S  /\  A  e.  S )  ->  ( B R A  <->  -.  ( B  =  A  \/  A R B ) ) )
76con2bid 320 . . . . . 6  |-  ( ( B  e.  S  /\  A  e.  S )  ->  ( ( B  =  A  \/  A R B )  <->  -.  B R A ) )
8 breq1 4207 . . . . . . . 8  |-  ( B  =  A  ->  ( B R C  <->  A R C ) )
98biimpd 199 . . . . . . 7  |-  ( B  =  A  ->  ( B R C  ->  A R C ) )
104, 1sotri 5253 . . . . . . . 8  |-  ( ( A R B  /\  B R C )  ->  A R C )
1110ex 424 . . . . . . 7  |-  ( A R B  ->  ( B R C  ->  A R C ) )
129, 11jaoi 369 . . . . . 6  |-  ( ( B  =  A  \/  A R B )  -> 
( B R C  ->  A R C ) )
137, 12syl6bir 221 . . . . 5  |-  ( ( B  e.  S  /\  A  e.  S )  ->  ( -.  B R A  ->  ( B R C  ->  A R C ) ) )
1413com3r 75 . . . 4  |-  ( B R C  ->  (
( B  e.  S  /\  A  e.  S
)  ->  ( -.  B R A  ->  A R C ) ) )
153, 14mpand 657 . . 3  |-  ( B R C  ->  ( A  e.  S  ->  ( -.  B R A  ->  A R C ) ) )
1615com3l 77 . 2  |-  ( A  e.  S  ->  ( -.  B R A  -> 
( B R C  ->  A R C ) ) )
17163imp 1147 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    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312   class class class wbr 4204    Or wor 4494    X. cxp 4868
This theorem is referenced by:  supsrlem  8978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-po 4495  df-so 4496  df-xp 4876
  Copyright terms: Public domain W3C validator