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Theorem solin 4829
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 4456 . . . . 5  |-  ( x  =  B  ->  (
x R y  <->  B R
y ) )
2 eqeq1 2471 . . . . 5  |-  ( x  =  B  ->  (
x  =  y  <->  B  =  y ) )
3 breq2 4457 . . . . 5  |-  ( x  =  B  ->  (
y R x  <->  y R B ) )
41, 2, 33orbi123d 1298 . . . 4  |-  ( x  =  B  ->  (
( x R y  \/  x  =  y  \/  y R x )  <->  ( B R y  \/  B  =  y  \/  y R B ) ) )
54imbi2d 316 . . 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 4457 . . . . 5  |-  ( y  =  C  ->  ( B R y  <->  B R C ) )
7 eqeq2 2482 . . . . 5  |-  ( y  =  C  ->  ( B  =  y  <->  B  =  C ) )
8 breq1 4456 . . . . 5  |-  ( y  =  C  ->  (
y R B  <->  C R B ) )
96, 7, 83orbi123d 1298 . . . 4  |-  ( y  =  C  ->  (
( B R y  \/  B  =  y  \/  y R B )  <->  ( B R C  \/  B  =  C  \/  C R B ) ) )
109imbi2d 316 . . 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 4807 . . . . 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 2841 . . . . . 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 466 . . . . 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 195 . . . 4  |-  ( R  Or  A  ->  (
( x  e.  A  /\  y  e.  A
)  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
1514com12 31 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( R  Or  A  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
165, 10, 15vtocl2ga 3184 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( R  Or  A  ->  ( B R C  \/  B  =  C  \/  C R B ) ) )
1716impcom 430 1  |-  ( ( R  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B R C  \/  B  =  C  \/  C R B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767   A.wral 2817   class class class wbr 4453    Po wpo 4804    Or wor 4805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-so 4807
This theorem is referenced by:  sotric  4832  sotrieq  4833  somo  4840  wecmpep  4877  sorpssi  6581  soxp  6908  wemaplem2  7984  fpwwe2lem12  9031  fpwwe2lem13  9032  lttri4  9681  xmullem  11468  xmulasslem  11489  orngsqr  27619  socnv  29121  wfrlem10  29279  slttri  29360  fin2so  29967  fnwe2lem3  30926
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