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Theorem soinxp 4943
Description: Intersection of total order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
soinxp  |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A
) )  Or  A
)

Proof of Theorem soinxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poinxp 4942 . . 3  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
2 brinxp 4941 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
3 biidd 230 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x  =  y  <-> 
x  =  y ) )
4 brinxp 4941 . . . . . . 7  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
54ancoms 441 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
62, 3, 53orbi123d 1254 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( x R y  \/  x  =  y  \/  y R x )  <->  ( x
( R  i^i  ( A  X.  A ) ) y  \/  x  =  y  \/  y ( R  i^i  ( A  X.  A ) ) x ) ) )
76ralbidva 2722 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x )  <->  A. y  e.  A  ( x ( R  i^i  ( A  X.  A ) ) y  \/  x  =  y  \/  y ( R  i^i  ( A  X.  A ) ) x ) ) )
87ralbiia 2738 . . 3  |-  ( A. x  e.  A  A. y  e.  A  (
x R y  \/  x  =  y  \/  y R x )  <->  A. x  e.  A  A. y  e.  A  ( x ( R  i^i  ( A  X.  A ) ) y  \/  x  =  y  \/  y ( R  i^i  ( A  X.  A ) ) x ) )
91, 8anbi12i 680 . 2  |-  ( ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  (
x R y  \/  x  =  y  \/  y R x ) )  <->  ( ( R  i^i  ( A  X.  A ) )  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x
( R  i^i  ( A  X.  A ) ) y  \/  x  =  y  \/  y ( R  i^i  ( A  X.  A ) ) x ) ) )
10 df-so 4505 . 2  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) ) )
11 df-so 4505 . 2  |-  ( ( R  i^i  ( A  X.  A ) )  Or  A  <->  ( ( R  i^i  ( A  X.  A ) )  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x
( R  i^i  ( A  X.  A ) ) y  \/  x  =  y  \/  y ( R  i^i  ( A  X.  A ) ) x ) ) )
129, 10, 113bitr4i 270 1  |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A
) )  Or  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    \/ w3o 936    e. wcel 1726   A.wral 2706    i^i cin 3320   class class class wbr 4213    Po wpo 4502    Or wor 4503    X. cxp 4877
This theorem is referenced by:  weinxp  4946  ltsopi  8766  cnso  12847  opsrtoslem2  16546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-po 4504  df-so 4505  df-xp 4885
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