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Theorem soinxp 5053
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 5052 . . 3  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
2 brinxp 5051 . . . . . 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 5051 . . . . . . 7  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
54ancoms 451 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
62, 3, 53orbi123d 1296 . . . . 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 2890 . . . 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 2884 . . 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 695 . 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 4790 . 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 4790 . 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 367    \/ w3o 970    e. wcel 1823   A.wral 2804    i^i cin 3460   class class class wbr 4439    Po wpo 4787    Or wor 4788    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-po 4789  df-so 4790  df-xp 4994
This theorem is referenced by:  weinxp  5056  ltsopi  9255  cnso  14067  opsrtoslem2  18347
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