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Theorem soinxp 4770
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 4769 . . 3  |-  ( R  Po  A  <->  ( R  i^i  ( A  X.  A
) )  Po  A
)
2 brinxp 4768 . . . . . 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 4768 . . . . . . 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 2572 . . . 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 2588 . . 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 4331 . 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 4331 . 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 1632    e. wcel 1696   A.wral 2556    i^i cin 3164   class class class wbr 4039    Po wpo 4328    Or wor 4329    X. cxp 4703
This theorem is referenced by:  weinxp  4773  ltsopi  8528  cnso  12541  opsrtoslem2  16242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-po 4330  df-so 4331  df-xp 4711
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