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Theorem soxp 6488
Description: A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
Hypothesis
Ref Expression
soxp.1  |-  T  =  { <. x ,  y
>.  |  ( (
x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x
) R ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x ) S ( 2nd `  y
) ) ) ) }
Assertion
Ref Expression
soxp  |-  ( ( R  Or  A  /\  S  Or  B )  ->  T  Or  ( A  X.  B ) )
Distinct variable groups:    x, A, y    x, B, y    x, R, y    x, S, y
Allowed substitution hints:    T( x, y)

Proof of Theorem soxp
Dummy variables  a 
b  c  d  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sopo 4549 . . 3  |-  ( R  Or  A  ->  R  Po  A )
2 sopo 4549 . . 3  |-  ( S  Or  B  ->  S  Po  B )
3 soxp.1 . . . 4  |-  T  =  { <. x ,  y
>.  |  ( (
x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x
) R ( 1st `  y )  \/  (
( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x ) S ( 2nd `  y
) ) ) ) }
43poxp 6487 . . 3  |-  ( ( R  Po  A  /\  S  Po  B )  ->  T  Po  ( A  X.  B ) )
51, 2, 4syl2an 465 . 2  |-  ( ( R  Or  A  /\  S  Or  B )  ->  T  Po  ( A  X.  B ) )
6 elxp 4924 . . . . 5  |-  ( t  e.  ( A  X.  B )  <->  E. a E. b ( t  = 
<. a ,  b >.  /\  ( a  e.  A  /\  b  e.  B
) ) )
7 elxp 4924 . . . . 5  |-  ( u  e.  ( A  X.  B )  <->  E. c E. d ( u  = 
<. c ,  d >.  /\  ( c  e.  A  /\  d  e.  B
) ) )
8 ioran 478 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  ( ( a R c  \/  ( a  =  c  /\  b S d ) )  \/  ( a  =  c  /\  b  =  d ) )  <->  ( -.  ( a R c  \/  ( a  =  c  /\  b S d ) )  /\  -.  ( a  =  c  /\  b  =  d ) ) )
9 ioran 478 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  ( a R c  \/  ( a  =  c  /\  b S d ) )  <->  ( -.  a R c  /\  -.  ( a  =  c  /\  b S d ) ) )
10 ianor 476 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  ( a  =  c  /\  b S d )  <->  ( -.  a  =  c  \/  -.  b S d ) )
1110anbi2i 677 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( -.  a R c  /\  -.  ( a  =  c  /\  b S d ) )  <-> 
( -.  a R c  /\  ( -.  a  =  c  \/ 
-.  b S d ) ) )
129, 11bitri 242 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  ( a R c  \/  ( a  =  c  /\  b S d ) )  <->  ( -.  a R c  /\  ( -.  a  =  c  \/  -.  b S d ) ) )
13 ianor 476 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  ( a  =  c  /\  b  =  d )  <->  ( -.  a  =  c  \/  -.  b  =  d )
)
1412, 13anbi12i 680 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -.  ( a R c  \/  ( a  =  c  /\  b S d ) )  /\  -.  ( a  =  c  /\  b  =  d ) )  <-> 
( ( -.  a R c  /\  ( -.  a  =  c  \/  -.  b S d ) )  /\  ( -.  a  =  c  \/  -.  b  =  d ) ) )
158, 14bitri 242 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  ( ( a R c  \/  ( a  =  c  /\  b S d ) )  \/  ( a  =  c  /\  b  =  d ) )  <->  ( ( -.  a R c  /\  ( -.  a  =  c  \/  -.  b S d ) )  /\  ( -.  a  =  c  \/  -.  b  =  d )
) )
16 solin 4555 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  ->  (
a R c  \/  a  =  c  \/  c R a ) )
17 3orass 940 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a R c  \/  a  =  c  \/  c R a )  <-> 
( a R c  \/  ( a  =  c  \/  c R a ) ) )
18 df-or 361 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a R c  \/  ( a  =  c  \/  c R a ) )  <->  ( -.  a R c  ->  (
a  =  c  \/  c R a ) ) )
1917, 18bitri 242 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( a R c  \/  a  =  c  \/  c R a )  <-> 
( -.  a R c  ->  ( a  =  c  \/  c R a ) ) )
2016, 19sylib 190 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  ->  ( -.  a R c  -> 
( a  =  c  \/  c R a ) ) )
21 solin 4555 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  (
b S d  \/  b  =  d  \/  d S b ) )
22 3orass 940 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( b S d  \/  b  =  d  \/  d S b )  <-> 
( b S d  \/  ( b  =  d  \/  d S b ) ) )
23 df-or 361 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( b S d  \/  ( b  =  d  \/  d S b ) )  <->  ( -.  b S d  ->  (
b  =  d  \/  d S b ) ) )
2422, 23bitri 242 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( b S d  \/  b  =  d  \/  d S b )  <-> 
( -.  b S d  ->  ( b  =  d  \/  d S b ) ) )
2521, 24sylib 190 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  ( -.  b S d  -> 
( b  =  d  \/  d S b ) ) )
2625orim2d 815 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  (
( -.  a  =  c  \/  -.  b S d )  -> 
( -.  a  =  c  \/  ( b  =  d  \/  d S b ) ) ) )
2720, 26im2anan9 810 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( -.  a R c  /\  ( -.  a  =  c  \/  -.  b S d ) )  ->  (
( a  =  c  \/  c R a )  /\  ( -.  a  =  c  \/  ( b  =  d  \/  d S b ) ) ) ) )
28 pm2.53 364 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a  =  c  \/  c R a )  ->  ( -.  a  =  c  ->  c R a ) )
29 orc 376 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( c R a  ->  (
c R a  \/  ( c  =  a  /\  d S b ) ) )
3028, 29syl6 32 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( a  =  c  \/  c R a )  ->  ( -.  a  =  c  ->  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )
3130adantr 453 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( a  =  c  \/  c R a )  /\  ( -.  a  =  c  \/  ( b  =  d  \/  d S b ) ) )  -> 
( -.  a  =  c  ->  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )
32 orel1 373 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( -.  b  =  d  -> 
( ( b  =  d  \/  d S b )  ->  d S b ) )
3332orim2d 815 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  b  =  d  -> 
( ( -.  a  =  c  \/  (
b  =  d  \/  d S b ) )  ->  ( -.  a  =  c  \/  d S b ) ) )
3433anim2d 550 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  b  =  d  -> 
( ( ( a  =  c  \/  c R a )  /\  ( -.  a  =  c  \/  ( b  =  d  \/  d S b ) ) )  ->  ( (
a  =  c  \/  c R a )  /\  ( -.  a  =  c  \/  d S b ) ) ) )
35 imor 403 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( a  =  c  -> 
d S b )  <-> 
( -.  a  =  c  \/  d S b ) )
3635biimpri 199 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( -.  a  =  c  \/  d S b )  ->  ( a  =  c  ->  d S b ) )
3736com12 30 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( a  =  c  ->  (
( -.  a  =  c  \/  d S b )  ->  d S b ) )
38 equcomi 1693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( a  =  c  ->  c  =  a )
3938anim1i 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( a  =  c  /\  d S b )  -> 
( c  =  a  /\  d S b ) )
4039olcd 384 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( a  =  c  /\  d S b )  -> 
( c R a  \/  ( c  =  a  /\  d S b ) ) )
4140ex 425 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( a  =  c  ->  (
d S b  -> 
( c R a  \/  ( c  =  a  /\  d S b ) ) ) )
4237, 41syld 43 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( a  =  c  ->  (
( -.  a  =  c  \/  d S b )  ->  (
c R a  \/  ( c  =  a  /\  d S b ) ) ) )
4329a1d 24 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( c R a  ->  (
( -.  a  =  c  \/  d S b )  ->  (
c R a  \/  ( c  =  a  /\  d S b ) ) ) )
4442, 43jaoi 370 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a  =  c  \/  c R a )  ->  ( ( -.  a  =  c  \/  d S b )  ->  ( c R a  \/  ( c  =  a  /\  d S b ) ) ) )
4544imp 420 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( a  =  c  \/  c R a )  /\  ( -.  a  =  c  \/  d S b ) )  ->  ( c R a  \/  (
c  =  a  /\  d S b ) ) )
4634, 45syl6com 34 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( a  =  c  \/  c R a )  /\  ( -.  a  =  c  \/  ( b  =  d  \/  d S b ) ) )  -> 
( -.  b  =  d  ->  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )
4731, 46jaod 371 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( a  =  c  \/  c R a )  /\  ( -.  a  =  c  \/  ( b  =  d  \/  d S b ) ) )  -> 
( ( -.  a  =  c  \/  -.  b  =  d )  ->  ( c R a  \/  ( c  =  a  /\  d S b ) ) ) )
4827, 47syl6 32 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( -.  a R c  /\  ( -.  a  =  c  \/  -.  b S d ) )  ->  (
( -.  a  =  c  \/  -.  b  =  d )  -> 
( c R a  \/  ( c  =  a  /\  d S b ) ) ) ) )
4948imp3a 422 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( ( -.  a R c  /\  ( -.  a  =  c  \/  -.  b S d ) )  /\  ( -.  a  =  c  \/  -.  b  =  d )
)  ->  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )
5015, 49syl5bi 210 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( -.  ( ( a R c  \/  ( a  =  c  /\  b S d ) )  \/  (
a  =  c  /\  b  =  d )
)  ->  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )
51 df-3or 938 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( a R c  \/  ( a  =  c  /\  b S d ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( c R a  \/  (
c  =  a  /\  d S b ) ) )  <->  ( ( ( a R c  \/  ( a  =  c  /\  b S d ) )  \/  (
a  =  c  /\  b  =  d )
)  \/  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )
52 df-or 361 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( a R c  \/  ( a  =  c  /\  b S d ) )  \/  ( a  =  c  /\  b  =  d ) )  \/  ( c R a  \/  ( c  =  a  /\  d S b ) ) )  <-> 
( -.  ( ( a R c  \/  ( a  =  c  /\  b S d ) )  \/  (
a  =  c  /\  b  =  d )
)  ->  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )
5351, 52bitri 242 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a R c  \/  ( a  =  c  /\  b S d ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( c R a  \/  (
c  =  a  /\  d S b ) ) )  <->  ( -.  (
( a R c  \/  ( a  =  c  /\  b S d ) )  \/  ( a  =  c  /\  b  =  d ) )  ->  (
c R a  \/  ( c  =  a  /\  d S b ) ) ) )
5450, 53sylibr 205 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( a R c  \/  ( a  =  c  /\  b S d ) )  \/  ( a  =  c  /\  b  =  d )  \/  (
c R a  \/  ( c  =  a  /\  d S b ) ) ) )
55 pm3.2 436 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( a  e.  A  /\  c  e.  A
)  /\  ( b  e.  B  /\  d  e.  B ) )  -> 
( ( a R c  \/  ( a  =  c  /\  b S d ) )  ->  ( ( ( a  e.  A  /\  c  e.  A )  /\  ( b  e.  B  /\  d  e.  B
) )  /\  (
a R c  \/  ( a  =  c  /\  b S d ) ) ) ) )
5655ad2ant2l 728 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( a R c  \/  ( a  =  c  /\  b S d ) )  ->  ( ( ( a  e.  A  /\  c  e.  A )  /\  ( b  e.  B  /\  d  e.  B
) )  /\  (
a R c  \/  ( a  =  c  /\  b S d ) ) ) ) )
57 idd 23 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( a  =  c  /\  b  =  d )  ->  (
a  =  c  /\  b  =  d )
) )
58 simpr 449 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  ->  (
a  e.  A  /\  c  e.  A )
)
5958ancomd 440 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  ->  (
c  e.  A  /\  a  e.  A )
)
60 simpr 449 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  (
b  e.  B  /\  d  e.  B )
)
6160ancomd 440 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  (
d  e.  B  /\  b  e.  B )
)
62 pm3.2 436 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( c  e.  A  /\  a  e.  A
)  /\  ( d  e.  B  /\  b  e.  B ) )  -> 
( ( c R a  \/  ( c  =  a  /\  d S b ) )  ->  ( ( ( c  e.  A  /\  a  e.  A )  /\  ( d  e.  B  /\  b  e.  B
) )  /\  (
c R a  \/  ( c  =  a  /\  d S b ) ) ) ) )
6359, 61, 62syl2an 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( c R a  \/  ( c  =  a  /\  d S b ) )  ->  ( ( ( c  e.  A  /\  a  e.  A )  /\  ( d  e.  B  /\  b  e.  B
) )  /\  (
c R a  \/  ( c  =  a  /\  d S b ) ) ) ) )
6456, 57, 633orim123d 1263 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( ( a R c  \/  (
a  =  c  /\  b S d ) )  \/  ( a  =  c  /\  b  =  d )  \/  (
c R a  \/  ( c  =  a  /\  d S b ) ) )  -> 
( ( ( ( a  e.  A  /\  c  e.  A )  /\  ( b  e.  B  /\  d  e.  B
) )  /\  (
a R c  \/  ( a  =  c  /\  b S d ) ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( ( ( c  e.  A  /\  a  e.  A
)  /\  ( d  e.  B  /\  b  e.  B ) )  /\  ( c R a  \/  ( c  =  a  /\  d S b ) ) ) ) ) )
6554, 64mpd 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  /\  ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( ( ( a  e.  A  /\  c  e.  A )  /\  ( b  e.  B  /\  d  e.  B
) )  /\  (
a R c  \/  ( a  =  c  /\  b S d ) ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( ( ( c  e.  A  /\  a  e.  A
)  /\  ( d  e.  B  /\  b  e.  B ) )  /\  ( c R a  \/  ( c  =  a  /\  d S b ) ) ) ) )
6665an4s 801 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  A  /\  c  e.  A )  /\  ( b  e.  B  /\  d  e.  B
) ) )  -> 
( ( ( ( a  e.  A  /\  c  e.  A )  /\  ( b  e.  B  /\  d  e.  B
) )  /\  (
a R c  \/  ( a  =  c  /\  b S d ) ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( ( ( c  e.  A  /\  a  e.  A
)  /\  ( d  e.  B  /\  b  e.  B ) )  /\  ( c R a  \/  ( c  =  a  /\  d S b ) ) ) ) )
6766expcom 426 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  A  /\  c  e.  A
)  /\  ( b  e.  B  /\  d  e.  B ) )  -> 
( ( R  Or  A  /\  S  Or  B
)  ->  ( (
( ( a  e.  A  /\  c  e.  A )  /\  (
b  e.  B  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( ( ( c  e.  A  /\  a  e.  A )  /\  (
d  e.  B  /\  b  e.  B )
)  /\  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) ) ) )
6867an4s 801 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  ( c  e.  A  /\  d  e.  B ) )  -> 
( ( R  Or  A  /\  S  Or  B
)  ->  ( (
( ( a  e.  A  /\  c  e.  A )  /\  (
b  e.  B  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( ( ( c  e.  A  /\  a  e.  A )  /\  (
d  e.  B  /\  b  e.  B )
)  /\  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) ) ) )
69 breq12 4242 . . . . . . . . . . . . . . . . 17  |-  ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( t T u  <->  <. a ,  b >. T <. c ,  d >. )
)
70 eqeq12 2454 . . . . . . . . . . . . . . . . 17  |-  ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( t  =  u  <->  <. a ,  b >.  =  <. c ,  d >. )
)
71 breq12 4242 . . . . . . . . . . . . . . . . . 18  |-  ( ( u  =  <. c ,  d >.  /\  t  =  <. a ,  b
>. )  ->  ( u T t  <->  <. c ,  d >. T <. a ,  b >. )
)
7271ancoms 441 . . . . . . . . . . . . . . . . 17  |-  ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( u T t  <->  <. c ,  d >. T <. a ,  b >. )
)
7369, 70, 723orbi123d 1254 . . . . . . . . . . . . . . . 16  |-  ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( ( t T u  \/  t  =  u  \/  u T t )  <-> 
( <. a ,  b
>. T <. c ,  d
>.  \/  <. a ,  b
>.  =  <. c ,  d >.  \/  <. c ,  d >. T <. a ,  b >. )
) )
743xporderlem 6486 . . . . . . . . . . . . . . . . 17  |-  ( <.
a ,  b >. T <. c ,  d
>. 
<->  ( ( ( a  e.  A  /\  c  e.  A )  /\  (
b  e.  B  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) ) )
75 vex 2965 . . . . . . . . . . . . . . . . . 18  |-  a  e. 
_V
76 vex 2965 . . . . . . . . . . . . . . . . . 18  |-  b  e. 
_V
7775, 76opth 4464 . . . . . . . . . . . . . . . . 17  |-  ( <.
a ,  b >.  =  <. c ,  d
>. 
<->  ( a  =  c  /\  b  =  d ) )
783xporderlem 6486 . . . . . . . . . . . . . . . . 17  |-  ( <.
c ,  d >. T <. a ,  b
>. 
<->  ( ( ( c  e.  A  /\  a  e.  A )  /\  (
d  e.  B  /\  b  e.  B )
)  /\  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )
7974, 77, 783orbi123i 1144 . . . . . . . . . . . . . . . 16  |-  ( (
<. a ,  b >. T <. c ,  d
>.  \/  <. a ,  b
>.  =  <. c ,  d >.  \/  <. c ,  d >. T <. a ,  b >. )  <->  ( ( ( ( a  e.  A  /\  c  e.  A )  /\  (
b  e.  B  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( ( ( c  e.  A  /\  a  e.  A )  /\  (
d  e.  B  /\  b  e.  B )
)  /\  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) ) )
8073, 79syl6bb 254 . . . . . . . . . . . . . . 15  |-  ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( ( t T u  \/  t  =  u  \/  u T t )  <-> 
( ( ( ( a  e.  A  /\  c  e.  A )  /\  ( b  e.  B  /\  d  e.  B
) )  /\  (
a R c  \/  ( a  =  c  /\  b S d ) ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( ( ( c  e.  A  /\  a  e.  A
)  /\  ( d  e.  B  /\  b  e.  B ) )  /\  ( c R a  \/  ( c  =  a  /\  d S b ) ) ) ) ) )
8180biimprcd 218 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( a  e.  A  /\  c  e.  A )  /\  (
b  e.  B  /\  d  e.  B )
)  /\  ( a R c  \/  (
a  =  c  /\  b S d ) ) )  \/  ( a  =  c  /\  b  =  d )  \/  ( ( ( c  e.  A  /\  a  e.  A )  /\  (
d  e.  B  /\  b  e.  B )
)  /\  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )  ->  (
( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( t T u  \/  t  =  u  \/  u T t ) ) )
8268, 81syl6 32 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  ( c  e.  A  /\  d  e.  B ) )  -> 
( ( R  Or  A  /\  S  Or  B
)  ->  ( (
t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( t T u  \/  t  =  u  \/  u T t ) ) ) )
8382com3r 76 . . . . . . . . . . . 12  |-  ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( ( ( a  e.  A  /\  b  e.  B
)  /\  ( c  e.  A  /\  d  e.  B ) )  -> 
( ( R  Or  A  /\  S  Or  B
)  ->  ( t T u  \/  t  =  u  \/  u T t ) ) ) )
8483imp 420 . . . . . . . . . . 11  |-  ( ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  /\  (
( a  e.  A  /\  b  e.  B
)  /\  ( c  e.  A  /\  d  e.  B ) ) )  ->  ( ( R  Or  A  /\  S  Or  B )  ->  (
t T u  \/  t  =  u  \/  u T t ) ) )
8584an4s 801 . . . . . . . . . 10  |-  ( ( ( t  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  /\  ( u  =  <. c ,  d
>.  /\  ( c  e.  A  /\  d  e.  B ) ) )  ->  ( ( R  Or  A  /\  S  Or  B )  ->  (
t T u  \/  t  =  u  \/  u T t ) ) )
8685expcom 426 . . . . . . . . 9  |-  ( ( u  =  <. c ,  d >.  /\  (
c  e.  A  /\  d  e.  B )
)  ->  ( (
t  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  ->  ( ( R  Or  A  /\  S  Or  B )  ->  ( t T u  \/  t  =  u  \/  u T t ) ) ) )
8786exlimivv 1646 . . . . . . . 8  |-  ( E. c E. d ( u  =  <. c ,  d >.  /\  (
c  e.  A  /\  d  e.  B )
)  ->  ( (
t  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  ->  ( ( R  Or  A  /\  S  Or  B )  ->  ( t T u  \/  t  =  u  \/  u T t ) ) ) )
8887com12 30 . . . . . . 7  |-  ( ( t  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  ->  ( E. c E. d ( u  =  <. c ,  d
>.  /\  ( c  e.  A  /\  d  e.  B ) )  -> 
( ( R  Or  A  /\  S  Or  B
)  ->  ( t T u  \/  t  =  u  \/  u T t ) ) ) )
8988exlimivv 1646 . . . . . 6  |-  ( E. a E. b ( t  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  ->  ( E. c E. d ( u  =  <. c ,  d
>.  /\  ( c  e.  A  /\  d  e.  B ) )  -> 
( ( R  Or  A  /\  S  Or  B
)  ->  ( t T u  \/  t  =  u  \/  u T t ) ) ) )
9089imp 420 . . . . 5  |-  ( ( E. a E. b
( t  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  /\  E. c E. d ( u  = 
<. c ,  d >.  /\  ( c  e.  A  /\  d  e.  B
) ) )  -> 
( ( R  Or  A  /\  S  Or  B
)  ->  ( t T u  \/  t  =  u  \/  u T t ) ) )
916, 7, 90syl2anb 467 . . . 4  |-  ( ( t  e.  ( A  X.  B )  /\  u  e.  ( A  X.  B ) )  -> 
( ( R  Or  A  /\  S  Or  B
)  ->  ( t T u  \/  t  =  u  \/  u T t ) ) )
9291com12 30 . . 3  |-  ( ( R  Or  A  /\  S  Or  B )  ->  ( ( t  e.  ( A  X.  B
)  /\  u  e.  ( A  X.  B
) )  ->  (
t T u  \/  t  =  u  \/  u T t ) ) )
9392ralrimivv 2803 . 2  |-  ( ( R  Or  A  /\  S  Or  B )  ->  A. t  e.  ( A  X.  B ) A. u  e.  ( A  X.  B ) ( t T u  \/  t  =  u  \/  u T t ) )
94 df-so 4533 . 2  |-  ( T  Or  ( A  X.  B )  <->  ( T  Po  ( A  X.  B
)  /\  A. t  e.  ( A  X.  B
) A. u  e.  ( A  X.  B
) ( t T u  \/  t  =  u  \/  u T t ) ) )
955, 93, 94sylanbrc 647 1  |-  ( ( R  Or  A  /\  S  Or  B )  ->  T  Or  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 936   E.wex 1551    = wceq 1653    e. wcel 1727   A.wral 2711   <.cop 3841   class class class wbr 4237   {copab 4290    Po wpo 4530    Or wor 4531    X. cxp 4905   ` cfv 5483   1stc1st 6376   2ndc2nd 6377
This theorem is referenced by:  wexp  6489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
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-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fv 5491  df-1st 6378  df-2nd 6379
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