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Theorem soinxp 5006
Description: Intersection of total order with Cartesian 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 5005 . . 3  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
2 brinxp 5004 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
3 biidd 237 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x  =  y  <-> 
x  =  y ) )
4 brinxp 5004 . . . . . . 7  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
54ancoms 453 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
62, 3, 53orbi123d 1289 . . . . 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 2841 . . . 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 2835 . . 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 697 . 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 4745 . 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 4745 . 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 277 1  |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A
) )  Or  A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    \/ w3o 964    e. wcel 1758   A.wral 2796    i^i cin 3430   class class class wbr 4395    Po wpo 4742    Or wor 4743    X. cxp 4941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-po 4744  df-so 4745  df-xp 4949
This theorem is referenced by:  weinxp  5009  ltsopi  9163  cnso  13642  opsrtoslem2  17685
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