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Theorem solin 4518
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 4207 . . . . 5  |-  ( x  =  B  ->  (
x R y  <->  B R
y ) )
2 eqeq1 2441 . . . . 5  |-  ( x  =  B  ->  (
x  =  y  <->  B  =  y ) )
3 breq2 4208 . . . . 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 4208 . . . . 5  |-  ( y  =  C  ->  ( B R y  <->  B R C ) )
7 eqeq2 2444 . . . . 5  |-  ( y  =  C  ->  ( B  =  y  <->  B  =  C ) )
8 breq1 4207 . . . . 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 4496 . . . . 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 2760 . . . . . 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 3011 . 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 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204    Po wpo 4493    Or wor 4494
This theorem is referenced by:  sotric  4521  sotrieq  4522  somo  4529  wecmpep  4566  soxp  6451  sorpssi  6520  wemaplem2  7508  fpwwe2lem12  8508  fpwwe2lem13  8509  lttri4  9151  xmullem  10835  xmulasslem  10856  ofldsqr  24232  socnv  25380  wfrlem10  25539  slttri  25620  fnwe2lem3  27118
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-ral 2702  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-so 4496
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