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Theorem disjxpin 27661
Description: Derive a disjunction over a Cartesian product from the disjunctions over its first and second elements. (Contributed by Thierry Arnoux, 9-Mar-2018.)
Hypotheses
Ref Expression
disjxpin.1  |-  ( x  =  ( 1st `  p
)  ->  C  =  E )
disjxpin.2  |-  ( y  =  ( 2nd `  p
)  ->  D  =  F )
disjxpin.3  |-  ( ph  -> Disj  x  e.  A  C
)
disjxpin.4  |-  ( ph  -> Disj  y  e.  B  D
)
Assertion
Ref Expression
disjxpin  |-  ( ph  -> Disj  p  e.  ( A  X.  B ) ( E  i^i  F ) )
Distinct variable groups:    x, p, A    y, p, B    C, p    D, p    x, E   
y, F
Allowed substitution hints:    ph( x, y, p)    A( y)    B( x)    C( x, y)    D( x, y)    E( y, p)    F( x, p)

Proof of Theorem disjxpin
Dummy variables  a 
c  q  r  b  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 6803 . . . . . . . . 9  |-  ( q  e.  ( A  X.  B )  ->  ( 1st `  q )  e.  A )
21ad2antrl 725 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( 1st `  q
)  e.  A )
3 xp1st 6803 . . . . . . . . 9  |-  ( r  e.  ( A  X.  B )  ->  ( 1st `  r )  e.  A )
43ad2antll 726 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( 1st `  r
)  e.  A )
5 simpl 455 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  ->  ph )
6 disjxpin.3 . . . . . . . . . . 11  |-  ( ph  -> Disj  x  e.  A  C
)
7 disjors 4425 . . . . . . . . . . 11  |-  (Disj  x  e.  A  C  <->  A. a  e.  A  A. c  e.  A  ( a  =  c  \/  ( [_ a  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/) ) )
86, 7sylib 196 . . . . . . . . . 10  |-  ( ph  ->  A. a  e.  A  A. c  e.  A  ( a  =  c  \/  ( [_ a  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/) ) )
9 eqeq1 2458 . . . . . . . . . . . 12  |-  ( a  =  ( 1st `  q
)  ->  ( a  =  c  <->  ( 1st `  q
)  =  c ) )
10 csbeq1 3423 . . . . . . . . . . . . . 14  |-  ( a  =  ( 1st `  q
)  ->  [_ a  /  x ]_ C  =  [_ ( 1st `  q )  /  x ]_ C
)
1110ineq1d 3685 . . . . . . . . . . . . 13  |-  ( a  =  ( 1st `  q
)  ->  ( [_ a  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (
[_ ( 1st `  q
)  /  x ]_ C  i^i  [_ c  /  x ]_ C ) )
1211eqeq1d 2456 . . . . . . . . . . . 12  |-  ( a  =  ( 1st `  q
)  ->  ( ( [_ a  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/)  <->  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/) ) )
139, 12orbi12d 707 . . . . . . . . . . 11  |-  ( a  =  ( 1st `  q
)  ->  ( (
a  =  c  \/  ( [_ a  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/) )  <->  ( ( 1st `  q )  =  c  \/  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/) ) ) )
14 eqeq2 2469 . . . . . . . . . . . 12  |-  ( c  =  ( 1st `  r
)  ->  ( ( 1st `  q )  =  c  <->  ( 1st `  q
)  =  ( 1st `  r ) ) )
15 csbeq1 3423 . . . . . . . . . . . . . 14  |-  ( c  =  ( 1st `  r
)  ->  [_ c  /  x ]_ C  =  [_ ( 1st `  r )  /  x ]_ C
)
1615ineq2d 3686 . . . . . . . . . . . . 13  |-  ( c  =  ( 1st `  r
)  ->  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (
[_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C ) )
1716eqeq1d 2456 . . . . . . . . . . . 12  |-  ( c  =  ( 1st `  r
)  ->  ( ( [_ ( 1st `  q
)  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/)  <->  ( [_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/) ) )
1814, 17orbi12d 707 . . . . . . . . . . 11  |-  ( c  =  ( 1st `  r
)  ->  ( (
( 1st `  q
)  =  c  \/  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/) )  <->  ( ( 1st `  q )  =  ( 1st `  r
)  \/  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/) ) ) )
1913, 18rspc2v 3216 . . . . . . . . . 10  |-  ( ( ( 1st `  q
)  e.  A  /\  ( 1st `  r )  e.  A )  -> 
( A. a  e.  A  A. c  e.  A  ( a  =  c  \/  ( [_ a  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/) )  ->  ( ( 1st `  q )  =  ( 1st `  r )  \/  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/) ) ) )
208, 19syl5 32 . . . . . . . . 9  |-  ( ( ( 1st `  q
)  e.  A  /\  ( 1st `  r )  e.  A )  -> 
( ph  ->  ( ( 1st `  q )  =  ( 1st `  r
)  \/  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/) ) ) )
2120imp 427 . . . . . . . 8  |-  ( ( ( ( 1st `  q
)  e.  A  /\  ( 1st `  r )  e.  A )  /\  ph )  ->  ( ( 1st `  q )  =  ( 1st `  r
)  \/  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/) ) )
222, 4, 5, 21syl21anc 1225 . . . . . . 7  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( 1st `  q
)  =  ( 1st `  r )  \/  ( [_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/) ) )
23 xp2nd 6804 . . . . . . . . 9  |-  ( q  e.  ( A  X.  B )  ->  ( 2nd `  q )  e.  B )
2423ad2antrl 725 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( 2nd `  q
)  e.  B )
25 xp2nd 6804 . . . . . . . . 9  |-  ( r  e.  ( A  X.  B )  ->  ( 2nd `  r )  e.  B )
2625ad2antll 726 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( 2nd `  r
)  e.  B )
27 disjxpin.4 . . . . . . . . . . 11  |-  ( ph  -> Disj  y  e.  B  D
)
28 disjors 4425 . . . . . . . . . . 11  |-  (Disj  y  e.  B  D  <->  A. b  e.  B  A. d  e.  B  ( b  =  d  \/  ( [_ b  /  y ]_ D  i^i  [_ d  /  y ]_ D
)  =  (/) ) )
2927, 28sylib 196 . . . . . . . . . 10  |-  ( ph  ->  A. b  e.  B  A. d  e.  B  ( b  =  d  \/  ( [_ b  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (/) ) )
30 eqeq1 2458 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  q
)  ->  ( b  =  d  <->  ( 2nd `  q
)  =  d ) )
31 csbeq1 3423 . . . . . . . . . . . . . 14  |-  ( b  =  ( 2nd `  q
)  ->  [_ b  / 
y ]_ D  =  [_ ( 2nd `  q )  /  y ]_ D
)
3231ineq1d 3685 . . . . . . . . . . . . 13  |-  ( b  =  ( 2nd `  q
)  ->  ( [_ b  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (
[_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ d  /  y ]_ D ) )
3332eqeq1d 2456 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  q
)  ->  ( ( [_ b  /  y ]_ D  i^i  [_ d  /  y ]_ D
)  =  (/)  <->  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (/) ) )
3430, 33orbi12d 707 . . . . . . . . . . 11  |-  ( b  =  ( 2nd `  q
)  ->  ( (
b  =  d  \/  ( [_ b  / 
y ]_ D  i^i  [_ d  /  y ]_ D
)  =  (/) )  <->  ( ( 2nd `  q )  =  d  \/  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (/) ) ) )
35 eqeq2 2469 . . . . . . . . . . . 12  |-  ( d  =  ( 2nd `  r
)  ->  ( ( 2nd `  q )  =  d  <->  ( 2nd `  q
)  =  ( 2nd `  r ) ) )
36 csbeq1 3423 . . . . . . . . . . . . . 14  |-  ( d  =  ( 2nd `  r
)  ->  [_ d  / 
y ]_ D  =  [_ ( 2nd `  r )  /  y ]_ D
)
3736ineq2d 3686 . . . . . . . . . . . . 13  |-  ( d  =  ( 2nd `  r
)  ->  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (
[_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D ) )
3837eqeq1d 2456 . . . . . . . . . . . 12  |-  ( d  =  ( 2nd `  r
)  ->  ( ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (/)  <->  ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) ) )
3935, 38orbi12d 707 . . . . . . . . . . 11  |-  ( d  =  ( 2nd `  r
)  ->  ( (
( 2nd `  q
)  =  d  \/  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ d  /  y ]_ D
)  =  (/) )  <->  ( ( 2nd `  q )  =  ( 2nd `  r
)  \/  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) ) ) )
4034, 39rspc2v 3216 . . . . . . . . . 10  |-  ( ( ( 2nd `  q
)  e.  B  /\  ( 2nd `  r )  e.  B )  -> 
( A. b  e.  B  A. d  e.  B  ( b  =  d  \/  ( [_ b  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (/) )  ->  ( ( 2nd `  q )  =  ( 2nd `  r )  \/  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) ) )
4129, 40syl5 32 . . . . . . . . 9  |-  ( ( ( 2nd `  q
)  e.  B  /\  ( 2nd `  r )  e.  B )  -> 
( ph  ->  ( ( 2nd `  q )  =  ( 2nd `  r
)  \/  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) ) ) )
4241imp 427 . . . . . . . 8  |-  ( ( ( ( 2nd `  q
)  e.  B  /\  ( 2nd `  r )  e.  B )  /\  ph )  ->  ( ( 2nd `  q )  =  ( 2nd `  r
)  \/  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) ) )
4324, 26, 5, 42syl21anc 1225 . . . . . . 7  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( 2nd `  q
)  =  ( 2nd `  r )  \/  ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) ) )
4422, 43jca 530 . . . . . 6  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( 1st `  q )  =  ( 1st `  r )  \/  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/) )  /\  (
( 2nd `  q
)  =  ( 2nd `  r )  \/  ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) ) ) )
45 anddi 868 . . . . . 6  |-  ( ( ( ( 1st `  q
)  =  ( 1st `  r )  \/  ( [_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/) )  /\  ( ( 2nd `  q )  =  ( 2nd `  r )  \/  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) )  <->  ( (
( ( 1st `  q
)  =  ( 1st `  r )  /\  ( 2nd `  q )  =  ( 2nd `  r
) )  \/  (
( 1st `  q
)  =  ( 1st `  r )  /\  ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) ) )  \/  (
( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) ) ) )
4644, 45sylib 196 . . . . 5  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( ( 1st `  q )  =  ( 1st `  r
)  /\  ( 2nd `  q )  =  ( 2nd `  r ) )  \/  ( ( 1st `  q )  =  ( 1st `  r
)  /\  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) ) )  \/  (
( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) ) ) )
47 orass 522 . . . . 5  |-  ( ( ( ( ( 1st `  q )  =  ( 1st `  r )  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( 1st `  q
)  =  ( 1st `  r )  /\  ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) ) )  \/  (
( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) ) )  <->  ( (
( 1st `  q
)  =  ( 1st `  r )  /\  ( 2nd `  q )  =  ( 2nd `  r
) )  \/  (
( ( 1st `  q
)  =  ( 1st `  r )  /\  ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) )  \/  ( (
( [_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/)  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) ) ) ) )
4846, 47sylib 196 . . . 4  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( 1st `  q )  =  ( 1st `  r )  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( ( 1st `  q )  =  ( 1st `  r )  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) )  \/  (
( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) ) ) ) )
49 xpopth 6812 . . . . . . 7  |-  ( ( q  e.  ( A  X.  B )  /\  r  e.  ( A  X.  B ) )  -> 
( ( ( 1st `  q )  =  ( 1st `  r )  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  <->  q  =  r ) )
5049adantl 464 . . . . . 6  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( 1st `  q )  =  ( 1st `  r )  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  <->  q  =  r ) )
5150biimpd 207 . . . . 5  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( 1st `  q )  =  ( 1st `  r )  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  -> 
q  =  r ) )
52 inss2 3705 . . . . . . . . . 10  |-  ( (
[_ q  /  p ]_ E  i^i  [_ r  /  p ]_ E )  i^i  ( [_ q  /  p ]_ F  i^i  [_ r  /  p ]_ F ) )  C_  ( [_ q  /  p ]_ F  i^i  [_ r  /  p ]_ F )
53 csbin 3849 . . . . . . . . . . . 12  |-  [_ q  /  p ]_ ( E  i^i  F )  =  ( [_ q  /  p ]_ E  i^i  [_ q  /  p ]_ F )
54 csbin 3849 . . . . . . . . . . . 12  |-  [_ r  /  p ]_ ( E  i^i  F )  =  ( [_ r  /  p ]_ E  i^i  [_ r  /  p ]_ F )
5553, 54ineq12i 3684 . . . . . . . . . . 11  |-  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  ( ( [_ q  /  p ]_ E  i^i  [_ q  /  p ]_ F )  i^i  ( [_ r  /  p ]_ E  i^i  [_ r  /  p ]_ F ) )
56 in4 3700 . . . . . . . . . . 11  |-  ( (
[_ q  /  p ]_ E  i^i  [_ q  /  p ]_ F )  i^i  ( [_ r  /  p ]_ E  i^i  [_ r  /  p ]_ F ) )  =  ( ( [_ q  /  p ]_ E  i^i  [_ r  /  p ]_ E )  i^i  ( [_ q  /  p ]_ F  i^i  [_ r  /  p ]_ F ) )
5755, 56eqtri 2483 . . . . . . . . . 10  |-  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  ( ( [_ q  /  p ]_ E  i^i  [_ r  /  p ]_ E )  i^i  ( [_ q  /  p ]_ F  i^i  [_ r  /  p ]_ F ) )
58 vex 3109 . . . . . . . . . . . . 13  |-  q  e. 
_V
59 csbnestg 3837 . . . . . . . . . . . . 13  |-  ( q  e.  _V  ->  [_ q  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ [_ q  /  p ]_ ( 2nd `  p )  /  y ]_ D
)
6058, 59ax-mp 5 . . . . . . . . . . . 12  |-  [_ q  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ [_ q  /  p ]_ ( 2nd `  p )  /  y ]_ D
61 fvex 5858 . . . . . . . . . . . . . 14  |-  ( 2nd `  p )  e.  _V
62 disjxpin.2 . . . . . . . . . . . . . 14  |-  ( y  =  ( 2nd `  p
)  ->  D  =  F )
6361, 62csbie 3446 . . . . . . . . . . . . 13  |-  [_ ( 2nd `  p )  / 
y ]_ D  =  F
6463csbeq2i 3832 . . . . . . . . . . . 12  |-  [_ q  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ q  /  p ]_ F
65 csbfv 5885 . . . . . . . . . . . . 13  |-  [_ q  /  p ]_ ( 2nd `  p )  =  ( 2nd `  q )
66 csbeq1 3423 . . . . . . . . . . . . 13  |-  ( [_ q  /  p ]_ ( 2nd `  p )  =  ( 2nd `  q
)  ->  [_ [_ q  /  p ]_ ( 2nd `  p )  /  y ]_ D  =  [_ ( 2nd `  q )  / 
y ]_ D )
6765, 66ax-mp 5 . . . . . . . . . . . 12  |-  [_ [_ q  /  p ]_ ( 2nd `  p )  /  y ]_ D  =  [_ ( 2nd `  q )  / 
y ]_ D
6860, 64, 673eqtr3ri 2492 . . . . . . . . . . 11  |-  [_ ( 2nd `  q )  / 
y ]_ D  =  [_ q  /  p ]_ F
69 vex 3109 . . . . . . . . . . . . 13  |-  r  e. 
_V
70 csbnestg 3837 . . . . . . . . . . . . 13  |-  ( r  e.  _V  ->  [_ r  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ [_ r  /  p ]_ ( 2nd `  p )  /  y ]_ D
)
7169, 70ax-mp 5 . . . . . . . . . . . 12  |-  [_ r  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ [_ r  /  p ]_ ( 2nd `  p )  /  y ]_ D
7263csbeq2i 3832 . . . . . . . . . . . 12  |-  [_ r  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ r  /  p ]_ F
73 csbfv 5885 . . . . . . . . . . . . 13  |-  [_ r  /  p ]_ ( 2nd `  p )  =  ( 2nd `  r )
74 csbeq1 3423 . . . . . . . . . . . . 13  |-  ( [_ r  /  p ]_ ( 2nd `  p )  =  ( 2nd `  r
)  ->  [_ [_ r  /  p ]_ ( 2nd `  p )  /  y ]_ D  =  [_ ( 2nd `  r )  / 
y ]_ D )
7573, 74ax-mp 5 . . . . . . . . . . . 12  |-  [_ [_ r  /  p ]_ ( 2nd `  p )  /  y ]_ D  =  [_ ( 2nd `  r )  / 
y ]_ D
7671, 72, 753eqtr3ri 2492 . . . . . . . . . . 11  |-  [_ ( 2nd `  r )  / 
y ]_ D  =  [_ r  /  p ]_ F
7768, 76ineq12i 3684 . . . . . . . . . 10  |-  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (
[_ q  /  p ]_ F  i^i  [_ r  /  p ]_ F )
7852, 57, 773sstr4i 3528 . . . . . . . . 9  |-  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  C_  ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )
79 sseq0 3816 . . . . . . . . 9  |-  ( ( ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F ) ) 
C_  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  /\  ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) )  ->  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) )
8078, 79mpan 668 . . . . . . . 8  |-  ( (
[_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/)  ->  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F ) )  =  (/) )
8180a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/)  ->  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) ) )
8281adantld 465 . . . . . 6  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( 1st `  q )  =  ( 1st `  r )  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) )  ->  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) ) )
83 inss1 3704 . . . . . . . . . . 11  |-  ( (
[_ q  /  p ]_ E  i^i  [_ r  /  p ]_ E )  i^i  ( [_ q  /  p ]_ F  i^i  [_ r  /  p ]_ F ) )  C_  ( [_ q  /  p ]_ E  i^i  [_ r  /  p ]_ E )
84 csbnestg 3837 . . . . . . . . . . . . . 14  |-  ( q  e.  _V  ->  [_ q  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ [_ q  /  p ]_ ( 1st `  p )  /  x ]_ C
)
8558, 84ax-mp 5 . . . . . . . . . . . . 13  |-  [_ q  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ [_ q  /  p ]_ ( 1st `  p )  /  x ]_ C
86 fvex 5858 . . . . . . . . . . . . . . 15  |-  ( 1st `  p )  e.  _V
87 disjxpin.1 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  p
)  ->  C  =  E )
8886, 87csbie 3446 . . . . . . . . . . . . . 14  |-  [_ ( 1st `  p )  /  x ]_ C  =  E
8988csbeq2i 3832 . . . . . . . . . . . . 13  |-  [_ q  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ q  /  p ]_ E
90 csbfv 5885 . . . . . . . . . . . . . 14  |-  [_ q  /  p ]_ ( 1st `  p )  =  ( 1st `  q )
91 csbeq1 3423 . . . . . . . . . . . . . 14  |-  ( [_ q  /  p ]_ ( 1st `  p )  =  ( 1st `  q
)  ->  [_ [_ q  /  p ]_ ( 1st `  p )  /  x ]_ C  =  [_ ( 1st `  q )  /  x ]_ C )
9290, 91ax-mp 5 . . . . . . . . . . . . 13  |-  [_ [_ q  /  p ]_ ( 1st `  p )  /  x ]_ C  =  [_ ( 1st `  q )  /  x ]_ C
9385, 89, 923eqtr3ri 2492 . . . . . . . . . . . 12  |-  [_ ( 1st `  q )  /  x ]_ C  =  [_ q  /  p ]_ E
94 csbnestg 3837 . . . . . . . . . . . . . 14  |-  ( r  e.  _V  ->  [_ r  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ [_ r  /  p ]_ ( 1st `  p )  /  x ]_ C
)
9569, 94ax-mp 5 . . . . . . . . . . . . 13  |-  [_ r  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ [_ r  /  p ]_ ( 1st `  p )  /  x ]_ C
9688csbeq2i 3832 . . . . . . . . . . . . 13  |-  [_ r  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ r  /  p ]_ E
97 csbfv 5885 . . . . . . . . . . . . . 14  |-  [_ r  /  p ]_ ( 1st `  p )  =  ( 1st `  r )
98 csbeq1 3423 . . . . . . . . . . . . . 14  |-  ( [_ r  /  p ]_ ( 1st `  p )  =  ( 1st `  r
)  ->  [_ [_ r  /  p ]_ ( 1st `  p )  /  x ]_ C  =  [_ ( 1st `  r )  /  x ]_ C )
9997, 98ax-mp 5 . . . . . . . . . . . . 13  |-  [_ [_ r  /  p ]_ ( 1st `  p )  /  x ]_ C  =  [_ ( 1st `  r )  /  x ]_ C
10095, 96, 993eqtr3ri 2492 . . . . . . . . . . . 12  |-  [_ ( 1st `  r )  /  x ]_ C  =  [_ r  /  p ]_ E
10193, 100ineq12i 3684 . . . . . . . . . . 11  |-  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (
[_ q  /  p ]_ E  i^i  [_ r  /  p ]_ E )
10283, 57, 1013sstr4i 3528 . . . . . . . . . 10  |-  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  C_  ( [_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )
103 sseq0 3816 . . . . . . . . . 10  |-  ( ( ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F ) ) 
C_  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  /\  ( [_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/) )  ->  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) )
104102, 103mpan 668 . . . . . . . . 9  |-  ( (
[_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/)  ->  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F ) )  =  (/) )
105104a1i 11 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  ->  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) ) )
106105adantrd 466 . . . . . . 7  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/)  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  -> 
( [_ q  /  p ]_ ( E  i^i  F
)  i^i  [_ r  /  p ]_ ( E  i^i  F ) )  =  (/) ) )
10781adantld 465 . . . . . . 7  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/)  /\  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) )  ->  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) ) )
108106, 107jaod 378 . . . . . 6  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( (
[_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/)  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) )  ->  ( [_ q  /  p ]_ ( E  i^i  F
)  i^i  [_ r  /  p ]_ ( E  i^i  F ) )  =  (/) ) )
10982, 108jaod 378 . . . . 5  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( ( 1st `  q )  =  ( 1st `  r
)  /\  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) )  \/  ( (
( [_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/)  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) ) )  -> 
( [_ q  /  p ]_ ( E  i^i  F
)  i^i  [_ r  /  p ]_ ( E  i^i  F ) )  =  (/) ) )
11051, 109orim12d 836 . . . 4  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( ( ( ( 1st `  q )  =  ( 1st `  r
)  /\  ( 2nd `  q )  =  ( 2nd `  r ) )  \/  ( ( ( 1st `  q
)  =  ( 1st `  r )  /\  ( [_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (/) )  \/  ( (
( [_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (/)  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  \/  ( ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r )  /  x ]_ C )  =  (/)  /\  ( [_ ( 2nd `  q )  / 
y ]_ D  i^i  [_ ( 2nd `  r )  / 
y ]_ D )  =  (/) ) ) ) )  ->  ( q  =  r  \/  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) ) ) )
11148, 110mpd 15 . . 3  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( q  =  r  \/  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) ) )
112111ralrimivva 2875 . 2  |-  ( ph  ->  A. q  e.  ( A  X.  B ) A. r  e.  ( A  X.  B ) ( q  =  r  \/  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) ) )
113 disjors 4425 . 2  |-  (Disj  p  e.  ( A  X.  B
) ( E  i^i  F )  <->  A. q  e.  ( A  X.  B ) A. r  e.  ( A  X.  B ) ( q  =  r  \/  ( [_ q  /  p ]_ ( E  i^i  F )  i^i  [_ r  /  p ]_ ( E  i^i  F
) )  =  (/) ) )
114112, 113sylibr 212 1  |-  ( ph  -> Disj  p  e.  ( A  X.  B ) ( E  i^i  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   [_csb 3420    i^i cin 3460    C_ wss 3461   (/)c0 3783  Disj wdisj 4410    X. cxp 4986   ` cfv 5570   1stc1st 6771   2ndc2nd 6772
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-8 1825  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-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  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-uni 4236  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-1st 6773  df-2nd 6774
This theorem is referenced by:  sibfof  28549
  Copyright terms: Public domain W3C validator