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Theorem soinxp 4754
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 4753 . . 3  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
2 brinxp 4752 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <-> 
x ( R  i^i  ( A  X.  A
) ) y ) )
3 biidd 228 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x  =  y  <-> 
x  =  y ) )
4 brinxp 4752 . . . . . . 7  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
54ancoms 439 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
62, 3, 53orbi123d 1251 . . . . 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 2559 . . . 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 2575 . . 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 678 . 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 4315 . 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 4315 . 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 268 1  |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A
) )  Or  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151   class class class wbr 4023    Po wpo 4312    Or wor 4313    X. cxp 4687
This theorem is referenced by:  weinxp  4757  ltsopi  8512  cnso  12525  opsrtoslem2  16226
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-ne 2448  df-ral 2548  df-rex 2549  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-opab 4078  df-po 4314  df-so 4315  df-xp 4695
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