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Theorem soxp 6864
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 4734 . . 3  |-  ( R  Or  A  ->  R  Po  A )
2 sopo 4734 . . 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 6863 . . 3  |-  ( ( R  Po  A  /\  S  Po  B )  ->  T  Po  ( A  X.  B ) )
51, 2, 4syl2an 479 . 2  |-  ( ( R  Or  A  /\  S  Or  B )  ->  T  Po  ( A  X.  B ) )
6 elxp 4813 . . . . 5  |-  ( t  e.  ( A  X.  B )  <->  E. a E. b ( t  = 
<. a ,  b >.  /\  ( a  e.  A  /\  b  e.  B
) ) )
7 elxp 4813 . . . . 5  |-  ( u  e.  ( A  X.  B )  <->  E. c E. d ( u  = 
<. c ,  d >.  /\  ( c  e.  A  /\  d  e.  B
) ) )
8 ioran 492 . . . . . . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  ( a R c  \/  ( a  =  c  /\  b S d ) )  <->  ( -.  a R c  /\  -.  ( a  =  c  /\  b S d ) ) )
10 ianor 490 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  ( a  =  c  /\  b S d )  <->  ( -.  a  =  c  \/  -.  b S d ) )
1110anbi2i 698 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( -.  a R c  /\  -.  ( a  =  c  /\  b S d ) )  <-> 
( -.  a R c  /\  ( -.  a  =  c  \/ 
-.  b S d ) ) )
129, 11bitri 252 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  ( a R c  \/  ( a  =  c  /\  b S d ) )  <->  ( -.  a R c  /\  ( -.  a  =  c  \/  -.  b S d ) ) )
13 ianor 490 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  ( a  =  c  /\  b  =  d )  <->  ( -.  a  =  c  \/  -.  b  =  d )
)
1412, 13anbi12i 701 . . . . . . . . . . . . . . . . . . . . 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 252 . . . . . . . . . . . . . . . . . . . 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 4740 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  ->  (
a R c  \/  a  =  c  \/  c R a ) )
17 3orass 985 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a R c  \/  a  =  c  \/  c R a )  <-> 
( a R c  \/  ( a  =  c  \/  c R a ) ) )
18 df-or 371 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a R c  \/  ( a  =  c  \/  c R a ) )  <->  ( -.  a R c  ->  (
a  =  c  \/  c R a ) ) )
1917, 18bitri 252 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( a R c  \/  a  =  c  \/  c R a )  <-> 
( -.  a R c  ->  ( a  =  c  \/  c R a ) ) )
2016, 19sylib 199 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  ->  ( -.  a R c  -> 
( a  =  c  \/  c R a ) ) )
21 solin 4740 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  (
b S d  \/  b  =  d  \/  d S b ) )
22 3orass 985 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( b S d  \/  b  =  d  \/  d S b )  <-> 
( b S d  \/  ( b  =  d  \/  d S b ) ) )
23 df-or 371 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( b S d  \/  ( b  =  d  \/  d S b ) )  <->  ( -.  b S d  ->  (
b  =  d  \/  d S b ) ) )
2422, 23bitri 252 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( b S d  \/  b  =  d  \/  d S b )  <-> 
( -.  b S d  ->  ( b  =  d  \/  d S b ) ) )
2521, 24sylib 199 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  ( -.  b S d  -> 
( b  =  d  \/  d S b ) ) )
2625orim2d 848 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  (
( -.  a  =  c  \/  -.  b S d )  -> 
( -.  a  =  c  \/  ( b  =  d  \/  d S b ) ) ) )
2720, 26im2anan9 843 . . . . . . . . . . . . . . . . . . . . . 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 pm2.53 374 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a  =  c  \/  c R a )  ->  ( -.  a  =  c  ->  c R a ) )
29 orc 386 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( c R a  ->  (
c R a  \/  ( c  =  a  /\  d S b ) ) )
3028, 29syl6 34 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( a  =  c  \/  c R a )  ->  ( -.  a  =  c  ->  ( c R a  \/  (
c  =  a  /\  d S b ) ) ) )
3130adantr 466 . . . . . . . . . . . . . . . . . . . . . . 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 383 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( -.  b  =  d  -> 
( ( b  =  d  \/  d S b )  ->  d S b ) )
3332orim2d 848 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  b  =  d  -> 
( ( -.  a  =  c  \/  (
b  =  d  \/  d S b ) )  ->  ( -.  a  =  c  \/  d S b ) ) )
3433anim2d 567 . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( a  =  c  -> 
d S b )  <-> 
( -.  a  =  c  \/  d S b ) )
3635biimpri 209 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( -.  a  =  c  \/  d S b )  ->  ( a  =  c  ->  d S b ) )
3736com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( a  =  c  ->  (
( -.  a  =  c  \/  d S b )  ->  d S b ) )
38 equcomi 1847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( a  =  c  ->  c  =  a )
3938anim1i 570 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( a  =  c  /\  d S b )  -> 
( c  =  a  /\  d S b ) )
4039olcd 394 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( a  =  c  /\  d S b )  -> 
( c R a  \/  ( c  =  a  /\  d S b ) ) )
4140ex 435 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( a  =  c  ->  (
d S b  -> 
( c R a  \/  ( c  =  a  /\  d S b ) ) ) )
4237, 41syld 45 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( a  =  c  ->  (
( -.  a  =  c  \/  d S b )  ->  (
c R a  \/  ( c  =  a  /\  d S b ) ) ) )
4329a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( c R a  ->  (
( -.  a  =  c  \/  d S b )  ->  (
c R a  \/  ( c  =  a  /\  d S b ) ) ) )
4442, 43jaoi 380 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( a  =  c  \/  c R a )  ->  ( ( -.  a  =  c  \/  d S b )  ->  ( c R a  \/  ( c  =  a  /\  d S b ) ) ) )
4544imp 430 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( a  =  c  \/  c R a )  /\  ( -.  a  =  c  \/  d S b ) )  ->  ( c R a  \/  (
c  =  a  /\  d S b ) ) )
4634, 45syl6com 36 . . . . . . . . . . . . . . . . . . . . . . 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 381 . . . . . . . . . . . . . . . . . . . . . 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 34 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
4948impd 432 . . . . . . . . . . . . . . . . . . . 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 220 . . . . . . . . . . . . . . . . . . 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 983 . . . . . . . . . . . . . . . . . . . 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 371 . . . . . . . . . . . . . . . . . . . 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 252 . . . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . . . 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 pm3.2 448 . . . . . . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  ->  (
a  e.  A  /\  c  e.  A )
)
5958ancomd 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( R  Or  A  /\  ( a  e.  A  /\  c  e.  A
) )  ->  (
c  e.  A  /\  a  e.  A )
)
60 simpr 462 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  (
b  e.  B  /\  d  e.  B )
)
6160ancomd 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( S  Or  B  /\  ( b  e.  B  /\  d  e.  B
) )  ->  (
d  e.  B  /\  b  e.  B )
)
62 pm3.2 448 . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . 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 1343 . . . . . . . . . . . . . . . . . 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 833 . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . 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 833 . . . . . . . . . . . . . 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 4371 . . . . . . . . . . . . . . . . 17  |-  ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( t T u  <->  <. a ,  b >. T <. c ,  d >. )
)
70 eqeq12 2441 . . . . . . . . . . . . . . . . 17  |-  ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( t  =  u  <->  <. a ,  b >.  =  <. c ,  d >. )
)
71 breq12 4371 . . . . . . . . . . . . . . . . . 18  |-  ( ( u  =  <. c ,  d >.  /\  t  =  <. a ,  b
>. )  ->  ( u T t  <->  <. c ,  d >. T <. a ,  b >. )
)
7271ancoms 454 . . . . . . . . . . . . . . . . 17  |-  ( ( t  =  <. a ,  b >.  /\  u  =  <. c ,  d
>. )  ->  ( u T t  <->  <. c ,  d >. T <. a ,  b >. )
)
7369, 70, 723orbi123d 1334 . . . . . . . . . . . . . . . 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 6862 . . . . . . . . . . . . . . . . 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 3025 . . . . . . . . . . . . . . . . . 18  |-  a  e. 
_V
76 vex 3025 . . . . . . . . . . . . . . . . . 18  |-  b  e. 
_V
7775, 76opth 4638 . . . . . . . . . . . . . . . . 17  |-  ( <.
a ,  b >.  =  <. c ,  d
>. 
<->  ( a  =  c  /\  b  =  d ) )
783xporderlem 6862 . . . . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . . 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 264 . . . . . . . . . . . . . . 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 228 . . . . . . . . . . . . . 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 34 . . . . . . . . . . . . 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 82 . . . . . . . . . . . 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 430 . . . . . . . . . . 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 833 . . . . . . . . . 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 436 . . . . . . . . 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 1771 . . . . . . . 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 32 . . . . . . 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 1771 . . . . . 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 430 . . . . 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 481 . . . 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 32 . . 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 2785 . 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 4718 . 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 668 1  |-  ( ( R  Or  A  /\  S  Or  B )  ->  T  Or  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    = wceq 1437   E.wex 1657    e. wcel 1872   A.wral 2714   <.cop 3947   class class class wbr 4366   {copab 4424    Po wpo 4715    Or wor 4716    X. cxp 4794   ` cfv 5544   1stc1st 6749   2ndc2nd 6750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-iota 5508  df-fun 5546  df-fv 5552  df-1st 6751  df-2nd 6752
This theorem is referenced by:  wexp  6865
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