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Theorem disjxpin 25952
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 6627 . . . . . . . . 9  |-  ( q  e.  ( A  X.  B )  ->  ( 1st `  q )  e.  A )
21ad2antrl 727 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( 1st `  q
)  e.  A )
3 xp1st 6627 . . . . . . . . 9  |-  ( r  e.  ( A  X.  B )  ->  ( 1st `  r )  e.  A )
43ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( 1st `  r
)  e.  A )
5 simpl 457 . . . . . . . 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 4299 . . . . . . . . . . 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 2449 . . . . . . . . . . . 12  |-  ( a  =  ( 1st `  q
)  ->  ( a  =  c  <->  ( 1st `  q
)  =  c ) )
10 csbeq1 3312 . . . . . . . . . . . . . 14  |-  ( a  =  ( 1st `  q
)  ->  [_ a  /  x ]_ C  =  [_ ( 1st `  q )  /  x ]_ C
)
1110ineq1d 3572 . . . . . . . . . . . . 13  |-  ( a  =  ( 1st `  q
)  ->  ( [_ a  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (
[_ ( 1st `  q
)  /  x ]_ C  i^i  [_ c  /  x ]_ C ) )
1211eqeq1d 2451 . . . . . . . . . . . 12  |-  ( a  =  ( 1st `  q
)  ->  ( ( [_ a  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/)  <->  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (/) ) )
139, 12orbi12d 709 . . . . . . . . . . 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 2452 . . . . . . . . . . . 12  |-  ( c  =  ( 1st `  r
)  ->  ( ( 1st `  q )  =  c  <->  ( 1st `  q
)  =  ( 1st `  r ) ) )
15 csbeq1 3312 . . . . . . . . . . . . . 14  |-  ( c  =  ( 1st `  r
)  ->  [_ c  /  x ]_ C  =  [_ ( 1st `  r )  /  x ]_ C
)
1615ineq2d 3573 . . . . . . . . . . . . 13  |-  ( c  =  ( 1st `  r
)  ->  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ c  /  x ]_ C )  =  (
[_ ( 1st `  q
)  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C ) )
1716eqeq1d 2451 . . . . . . . . . . . 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 709 . . . . . . . . . . 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 3100 . . . . . . . . . 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 429 . . . . . . . 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 1217 . . . . . . 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 6628 . . . . . . . . 9  |-  ( q  e.  ( A  X.  B )  ->  ( 2nd `  q )  e.  B )
2423ad2antrl 727 . . . . . . . 8  |-  ( (
ph  /\  ( q  e.  ( A  X.  B
)  /\  r  e.  ( A  X.  B
) ) )  -> 
( 2nd `  q
)  e.  B )
25 xp2nd 6628 . . . . . . . . 9  |-  ( r  e.  ( A  X.  B )  ->  ( 2nd `  r )  e.  B )
2625ad2antll 728 . . . . . . . 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 4299 . . . . . . . . . . 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 2449 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  q
)  ->  ( b  =  d  <->  ( 2nd `  q
)  =  d ) )
31 csbeq1 3312 . . . . . . . . . . . . . 14  |-  ( b  =  ( 2nd `  q
)  ->  [_ b  / 
y ]_ D  =  [_ ( 2nd `  q )  /  y ]_ D
)
3231ineq1d 3572 . . . . . . . . . . . . 13  |-  ( b  =  ( 2nd `  q
)  ->  ( [_ b  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (
[_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ d  /  y ]_ D ) )
3332eqeq1d 2451 . . . . . . . . . . . 12  |-  ( b  =  ( 2nd `  q
)  ->  ( ( [_ b  /  y ]_ D  i^i  [_ d  /  y ]_ D
)  =  (/)  <->  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (/) ) )
3430, 33orbi12d 709 . . . . . . . . . . 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 2452 . . . . . . . . . . . 12  |-  ( d  =  ( 2nd `  r
)  ->  ( ( 2nd `  q )  =  d  <->  ( 2nd `  q
)  =  ( 2nd `  r ) ) )
36 csbeq1 3312 . . . . . . . . . . . . . 14  |-  ( d  =  ( 2nd `  r
)  ->  [_ d  / 
y ]_ D  =  [_ ( 2nd `  r )  /  y ]_ D
)
3736ineq2d 3573 . . . . . . . . . . . . 13  |-  ( d  =  ( 2nd `  r
)  ->  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ d  /  y ]_ D )  =  (
[_ ( 2nd `  q
)  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D ) )
3837eqeq1d 2451 . . . . . . . . . . . 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 709 . . . . . . . . . . 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 3100 . . . . . . . . . 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 429 . . . . . . . 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 1217 . . . . . . 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 532 . . . . . 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 865 . . . . . 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 524 . . . . 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 6636 . . . . . . 7  |-  ( ( q  e.  ( A  X.  B )  /\  r  e.  ( A  X.  B ) )  -> 
( ( ( 1st `  q )  =  ( 1st `  r )  /\  ( 2nd `  q
)  =  ( 2nd `  r ) )  <->  q  =  r ) )
5049adantl 466 . . . . . 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 3592 . . . . . . . . . 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 3733 . . . . . . . . . . . 12  |-  [_ q  /  p ]_ ( E  i^i  F )  =  ( [_ q  /  p ]_ E  i^i  [_ q  /  p ]_ F )
54 csbin 3733 . . . . . . . . . . . 12  |-  [_ r  /  p ]_ ( E  i^i  F )  =  ( [_ r  /  p ]_ E  i^i  [_ r  /  p ]_ F )
5553, 54ineq12i 3571 . . . . . . . . . . 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 3587 . . . . . . . . . . 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 2463 . . . . . . . . . 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 2996 . . . . . . . . . . . . 13  |-  q  e. 
_V
59 csbnestg 3715 . . . . . . . . . . . . 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 5722 . . . . . . . . . . . . . 14  |-  ( 2nd `  p )  e.  _V
62 nfcv 2589 . . . . . . . . . . . . . 14  |-  F/_ y F
63 disjxpin.2 . . . . . . . . . . . . . 14  |-  ( y  =  ( 2nd `  p
)  ->  D  =  F )
6461, 62, 63csbief 3334 . . . . . . . . . . . . 13  |-  [_ ( 2nd `  p )  / 
y ]_ D  =  F
6564csbeq2i 3709 . . . . . . . . . . . 12  |-  [_ q  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ q  /  p ]_ F
66 nfcv 2589 . . . . . . . . . . . . . 14  |-  F/_ p
( 2nd `  q
)
67 fveq2 5712 . . . . . . . . . . . . . 14  |-  ( p  =  q  ->  ( 2nd `  p )  =  ( 2nd `  q
) )
6858, 66, 67csbief 3334 . . . . . . . . . . . . 13  |-  [_ q  /  p ]_ ( 2nd `  p )  =  ( 2nd `  q )
69 csbeq1 3312 . . . . . . . . . . . . 13  |-  ( [_ q  /  p ]_ ( 2nd `  p )  =  ( 2nd `  q
)  ->  [_ [_ q  /  p ]_ ( 2nd `  p )  /  y ]_ D  =  [_ ( 2nd `  q )  / 
y ]_ D )
7068, 69ax-mp 5 . . . . . . . . . . . 12  |-  [_ [_ q  /  p ]_ ( 2nd `  p )  /  y ]_ D  =  [_ ( 2nd `  q )  / 
y ]_ D
7160, 65, 703eqtr3ri 2472 . . . . . . . . . . 11  |-  [_ ( 2nd `  q )  / 
y ]_ D  =  [_ q  /  p ]_ F
72 vex 2996 . . . . . . . . . . . . 13  |-  r  e. 
_V
73 csbnestg 3715 . . . . . . . . . . . . 13  |-  ( r  e.  _V  ->  [_ r  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ [_ r  /  p ]_ ( 2nd `  p )  /  y ]_ D
)
7472, 73ax-mp 5 . . . . . . . . . . . 12  |-  [_ r  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ [_ r  /  p ]_ ( 2nd `  p )  /  y ]_ D
7564csbeq2i 3709 . . . . . . . . . . . 12  |-  [_ r  /  p ]_ [_ ( 2nd `  p )  / 
y ]_ D  =  [_ r  /  p ]_ F
76 nfcv 2589 . . . . . . . . . . . . . 14  |-  F/_ p
( 2nd `  r
)
77 fveq2 5712 . . . . . . . . . . . . . 14  |-  ( p  =  r  ->  ( 2nd `  p )  =  ( 2nd `  r
) )
7872, 76, 77csbief 3334 . . . . . . . . . . . . 13  |-  [_ r  /  p ]_ ( 2nd `  p )  =  ( 2nd `  r )
79 csbeq1 3312 . . . . . . . . . . . . 13  |-  ( [_ r  /  p ]_ ( 2nd `  p )  =  ( 2nd `  r
)  ->  [_ [_ r  /  p ]_ ( 2nd `  p )  /  y ]_ D  =  [_ ( 2nd `  r )  / 
y ]_ D )
8078, 79ax-mp 5 . . . . . . . . . . . 12  |-  [_ [_ r  /  p ]_ ( 2nd `  p )  /  y ]_ D  =  [_ ( 2nd `  r )  / 
y ]_ D
8174, 75, 803eqtr3ri 2472 . . . . . . . . . . 11  |-  [_ ( 2nd `  r )  / 
y ]_ D  =  [_ r  /  p ]_ F
8271, 81ineq12i 3571 . . . . . . . . . 10  |-  ( [_ ( 2nd `  q )  /  y ]_ D  i^i  [_ ( 2nd `  r
)  /  y ]_ D )  =  (
[_ q  /  p ]_ F  i^i  [_ r  /  p ]_ F )
8352, 57, 823sstr4i 3416 . . . . . . . . 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 )
84 sseq0 3690 . . . . . . . . 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
) )  =  (/) )
8583, 84mpan 670 . . . . . . . 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 ) )  =  (/) )
8685a1i 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
) )  =  (/) ) )
8786adantld 467 . . . . . 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
) )  =  (/) ) )
88 inss1 3591 . . . . . . . . . . 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 )
89 csbnestg 3715 . . . . . . . . . . . . . 14  |-  ( q  e.  _V  ->  [_ q  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ [_ q  /  p ]_ ( 1st `  p )  /  x ]_ C
)
9058, 89ax-mp 5 . . . . . . . . . . . . 13  |-  [_ q  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ [_ q  /  p ]_ ( 1st `  p )  /  x ]_ C
91 fvex 5722 . . . . . . . . . . . . . . 15  |-  ( 1st `  p )  e.  _V
92 nfcv 2589 . . . . . . . . . . . . . . 15  |-  F/_ x E
93 disjxpin.1 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  p
)  ->  C  =  E )
9491, 92, 93csbief 3334 . . . . . . . . . . . . . 14  |-  [_ ( 1st `  p )  /  x ]_ C  =  E
9594csbeq2i 3709 . . . . . . . . . . . . 13  |-  [_ q  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ q  /  p ]_ E
96 nfcv 2589 . . . . . . . . . . . . . . 15  |-  F/_ p
( 1st `  q
)
97 fveq2 5712 . . . . . . . . . . . . . . 15  |-  ( p  =  q  ->  ( 1st `  p )  =  ( 1st `  q
) )
9858, 96, 97csbief 3334 . . . . . . . . . . . . . 14  |-  [_ q  /  p ]_ ( 1st `  p )  =  ( 1st `  q )
99 csbeq1 3312 . . . . . . . . . . . . . 14  |-  ( [_ q  /  p ]_ ( 1st `  p )  =  ( 1st `  q
)  ->  [_ [_ q  /  p ]_ ( 1st `  p )  /  x ]_ C  =  [_ ( 1st `  q )  /  x ]_ C )
10098, 99ax-mp 5 . . . . . . . . . . . . 13  |-  [_ [_ q  /  p ]_ ( 1st `  p )  /  x ]_ C  =  [_ ( 1st `  q )  /  x ]_ C
10190, 95, 1003eqtr3ri 2472 . . . . . . . . . . . 12  |-  [_ ( 1st `  q )  /  x ]_ C  =  [_ q  /  p ]_ E
102 csbnestg 3715 . . . . . . . . . . . . . 14  |-  ( r  e.  _V  ->  [_ r  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ [_ r  /  p ]_ ( 1st `  p )  /  x ]_ C
)
10372, 102ax-mp 5 . . . . . . . . . . . . 13  |-  [_ r  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ [_ r  /  p ]_ ( 1st `  p )  /  x ]_ C
10494csbeq2i 3709 . . . . . . . . . . . . 13  |-  [_ r  /  p ]_ [_ ( 1st `  p )  /  x ]_ C  =  [_ r  /  p ]_ E
105 nfcv 2589 . . . . . . . . . . . . . . 15  |-  F/_ p
( 1st `  r
)
106 fveq2 5712 . . . . . . . . . . . . . . 15  |-  ( p  =  r  ->  ( 1st `  p )  =  ( 1st `  r
) )
10772, 105, 106csbief 3334 . . . . . . . . . . . . . 14  |-  [_ r  /  p ]_ ( 1st `  p )  =  ( 1st `  r )
108 csbeq1 3312 . . . . . . . . . . . . . 14  |-  ( [_ r  /  p ]_ ( 1st `  p )  =  ( 1st `  r
)  ->  [_ [_ r  /  p ]_ ( 1st `  p )  /  x ]_ C  =  [_ ( 1st `  r )  /  x ]_ C )
109107, 108ax-mp 5 . . . . . . . . . . . . 13  |-  [_ [_ r  /  p ]_ ( 1st `  p )  /  x ]_ C  =  [_ ( 1st `  r )  /  x ]_ C
110103, 104, 1093eqtr3ri 2472 . . . . . . . . . . . 12  |-  [_ ( 1st `  r )  /  x ]_ C  =  [_ r  /  p ]_ E
111101, 110ineq12i 3571 . . . . . . . . . . 11  |-  ( [_ ( 1st `  q )  /  x ]_ C  i^i  [_ ( 1st `  r
)  /  x ]_ C )  =  (
[_ q  /  p ]_ E  i^i  [_ r  /  p ]_ E )
11288, 57, 1113sstr4i 3416 . . . . . . . . . 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 )
113 sseq0 3690 . . . . . . . . . 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
) )  =  (/) )
114112, 113mpan 670 . . . . . . . . 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 ) )  =  (/) )
115114a1i 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
) )  =  (/) ) )
116115adantrd 468 . . . . . . 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 ) )  =  (/) ) )
11786adantld 467 . . . . . . 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
) )  =  (/) ) )
118116, 117jaod 380 . . . . . 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 ) )  =  (/) ) )
11987, 118jaod 380 . . . . 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 ) )  =  (/) ) )
12051, 119orim12d 834 . . . 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
) )  =  (/) ) ) )
12148, 120mpd 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
) )  =  (/) ) )
122121ralrimivva 2829 . 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
) )  =  (/) ) )
123 disjors 4299 . 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
) )  =  (/) ) )
124122, 123sylibr 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 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   _Vcvv 2993   [_csb 3309    i^i cin 3348    C_ wss 3349   (/)c0 3658  Disj wdisj 4283    X. cxp 4859   ` cfv 5439   1stc1st 6596   2ndc2nd 6597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-disj 4284  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fv 5447  df-1st 6598  df-2nd 6599
This theorem is referenced by:  sibfof  26748
  Copyright terms: Public domain W3C validator