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Theorem solin 4337
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 4026 . . . . 5  |-  ( x  =  B  ->  (
x R y  <->  B R
y ) )
2 eqeq1 2289 . . . . 5  |-  ( x  =  B  ->  (
x  =  y  <->  B  =  y ) )
3 breq2 4027 . . . . 5  |-  ( x  =  B  ->  (
y R x  <->  y R B ) )
41, 2, 33orbi123d 1251 . . . 4  |-  ( x  =  B  ->  (
( x R y  \/  x  =  y  \/  y R x )  <->  ( B R y  \/  B  =  y  \/  y R B ) ) )
54imbi2d 307 . . 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 4027 . . . . 5  |-  ( y  =  C  ->  ( B R y  <->  B R C ) )
7 eqeq2 2292 . . . . 5  |-  ( y  =  C  ->  ( B  =  y  <->  B  =  C ) )
8 breq1 4026 . . . . 5  |-  ( y  =  C  ->  (
y R B  <->  C R B ) )
96, 7, 83orbi123d 1251 . . . 4  |-  ( y  =  C  ->  (
( B R y  \/  B  =  y  \/  y R B )  <->  ( B R C  \/  B  =  C  \/  C R B ) ) )
109imbi2d 307 . . 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 4315 . . . . 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 2605 . . . . . 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 452 . . . . 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 187 . . . 4  |-  ( R  Or  A  ->  (
( x  e.  A  /\  y  e.  A
)  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
1514com12 27 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( R  Or  A  ->  ( x R y  \/  x  =  y  \/  y R x ) ) )
165, 10, 15vtocl2ga 2851 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( R  Or  A  ->  ( B R C  \/  B  =  C  \/  C R B ) ) )
1716impcom 419 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 358    \/ w3o 933    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    Po wpo 4312    Or wor 4313
This theorem is referenced by:  sotric  4340  sotrieq  4341  somo  4348  wecmpep  4385  soxp  6228  sorpssi  6283  wemaplem2  7262  fpwwe2lem12  8263  fpwwe2lem13  8264  lttri4  8906  xmullem  10584  xmulasslem  10605  wfrlem10  24265  slttri  24327  fnwe2lem3  27149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-so 4315
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