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Theorem solin 4740
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 4369 . . . . 5  |-  ( x  =  B  ->  (
x R y  <->  B R
y ) )
2 eqeq1 2432 . . . . 5  |-  ( x  =  B  ->  (
x  =  y  <->  B  =  y ) )
3 breq2 4370 . . . . 5  |-  ( x  =  B  ->  (
y R x  <->  y R B ) )
41, 2, 33orbi123d 1334 . . . 4  |-  ( x  =  B  ->  (
( x R y  \/  x  =  y  \/  y R x )  <->  ( B R y  \/  B  =  y  \/  y R B ) ) )
54imbi2d 317 . . 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 4370 . . . . 5  |-  ( y  =  C  ->  ( B R y  <->  B R C ) )
7 eqeq2 2439 . . . . 5  |-  ( y  =  C  ->  ( B  =  y  <->  B  =  C ) )
8 breq1 4369 . . . . 5  |-  ( y  =  C  ->  (
y R B  <->  C R B ) )
96, 7, 83orbi123d 1334 . . . 4  |-  ( y  =  C  ->  (
( B R y  \/  B  =  y  \/  y R B )  <->  ( B R C  \/  B  =  C  \/  C R B ) ) )
109imbi2d 317 . . 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 4718 . . . . 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 2739 . . . . . 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 467 . . . . 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 198 . . . 4  |-  ( R  Or  A  ->  (
( x  e.  A  /\  y  e.  A
)  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
1514com12 32 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( R  Or  A  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
165, 10, 15vtocl2ga 3090 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( R  Or  A  ->  ( B R C  \/  B  =  C  \/  C R B ) ) )
1716impcom 431 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 370    \/ w3o 981    = wceq 1437    e. wcel 1872   A.wral 2714   class class class wbr 4366    Po wpo 4715    Or wor 4716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-so 4718
This theorem is referenced by:  sotric  4743  sotrieq  4744  somo  4751  wecmpep  4788  sorpssi  6535  soxp  6864  wfrlem10  7000  wemaplem2  8015  fpwwe2lem12  9017  fpwwe2lem13  9018  lttri4  9669  xmullem  11501  xmulasslem  11522  orngsqr  28519  socnv  30356  slttri  30511  fin2so  31839  fnwe2lem3  35823
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