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Theorem solin 4494
Description: A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.)
Assertion
Ref Expression
solin  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )

Proof of Theorem solin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4183 . . . . 5  |-  ( x  =  B  ->  (
x R y  <->  B R
y ) )
2 eqeq1 2418 . . . . 5  |-  ( x  =  B  ->  (
x  =  y  <->  B  =  y ) )
3 breq2 4184 . . . . 5  |-  ( x  =  B  ->  (
y R x  <->  y R B ) )
41, 2, 33orbi123d 1253 . . . 4  |-  ( x  =  B  ->  (
( x R y  \/  x  =  y  \/  y R x )  <->  ( B R y  \/  B  =  y  \/  y R B ) ) )
54imbi2d 308 . . 3  |-  ( x  =  B  ->  (
( R  Or  A  ->  ( x R y  \/  x  =  y  \/  y R x ) )  <->  ( R  Or  A  ->  ( B R y  \/  B  =  y  \/  y R B ) ) ) )
6 breq2 4184 . . . . 5  |-  ( y  =  C  ->  ( B R y  <->  B R C ) )
7 eqeq2 2421 . . . . 5  |-  ( y  =  C  ->  ( B  =  y  <->  B  =  C ) )
8 breq1 4183 . . . . 5  |-  ( y  =  C  ->  (
y R B  <->  C R B ) )
96, 7, 83orbi123d 1253 . . . 4  |-  ( y  =  C  ->  (
( B R y  \/  B  =  y  \/  y R B )  <->  ( B R C  \/  B  =  C  \/  C R B ) ) )
109imbi2d 308 . . 3  |-  ( y  =  C  ->  (
( R  Or  A  ->  ( B R y  \/  B  =  y  \/  y R B ) )  <->  ( R  Or  A  ->  ( B R C  \/  B  =  C  \/  C R B ) ) ) )
11 df-so 4472 . . . . 5  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) ) )
12 rsp2 2736 . . . . . 6  |-  ( A. x  e.  A  A. y  e.  A  (
x R y  \/  x  =  y  \/  y R x )  ->  ( ( x  e.  A  /\  y  e.  A )  ->  (
x R y  \/  x  =  y  \/  y R x ) ) )
1312adantl 453 . . . . 5  |-  ( ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  (
x R y  \/  x  =  y  \/  y R x ) )  ->  ( (
x  e.  A  /\  y  e.  A )  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
1411, 13sylbi 188 . . . 4  |-  ( R  Or  A  ->  (
( x  e.  A  /\  y  e.  A
)  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
1514com12 29 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( R  Or  A  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
165, 10, 15vtocl2ga 2987 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( R  Or  A  ->  ( B R C  \/  B  =  C  \/  C R B ) ) )
1716impcom 420 1  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1721   A.wral 2674   class class class wbr 4180    Po wpo 4469    Or wor 4470
This theorem is referenced by:  sotric  4497  sotrieq  4498  somo  4505  wecmpep  4542  soxp  6426  sorpssi  6495  wemaplem2  7480  fpwwe2lem12  8480  fpwwe2lem13  8481  lttri4  9123  xmullem  10807  xmulasslem  10828  ofldsqr  24201  wfrlem10  25487  slttri  25549  fnwe2lem3  27025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-so 4472
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