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Theorem disjxpin 29401
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 7198 . . . . . . . . 9 |- ( q e. ( A X. B ) -> ( 1st ` q ) e. A )
21ad2antrl 764 . . . . . . . 8 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 1st ` q ) e. A )
3 xp1st 7198 . . . . . . . . 9 |- ( r e. ( A X. B ) -> ( 1st ` r ) e. A )
43ad2antll 765 . . . . . . . 8 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 1st ` r ) e. A )
5 simpl 473 . . . . . . . 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 4635 . . . . . . . . . . 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 208 . . . . . . . . . 10 |- ( ph -> A. a e. A A. c e. A ( a = c \/ ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) ) )
9 eqeq1 2626 . . . . . . . . . . . 12 |- ( a = ( 1st ` q ) -> ( a = c <-> ( 1st ` q ) = c ) )
10 csbeq1 3536 . . . . . . . . . . . . . 14 |- ( a = ( 1st ` q ) -> [_ a / x ]_ C = [_ ( 1st ` q ) / x ]_ C )
1110ineq1d 3813 . . . . . . . . . . . . 13 |- ( a = ( 1st ` q ) -> ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) )
1211eqeq1d 2624 . . . . . . . . . . . 12 |- ( a = ( 1st ` q ) -> ( ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) <-> ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = (/) ) )
139, 12orbi12d 746 . . . . . . . . . . 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 2633 . . . . . . . . . . . 12 |- ( c = ( 1st ` r ) -> ( ( 1st ` q ) = c <-> ( 1st ` q ) = ( 1st ` r ) ) )
15 csbeq1 3536 . . . . . . . . . . . . . 14 |- ( c = ( 1st ` r ) -> [_ c / x ]_ C = [_ ( 1st ` r ) / x ]_ C )
1615ineq2d 3814 . . . . . . . . . . . . 13 |- ( c = ( 1st ` r ) -> ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) )
1716eqeq1d 2624 . . . . . . . . . . . 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 746 . . . . . . . . . . 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 3322 . . . . . . . . . 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 34 . . . . . . . . 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 445 . . . . . . . 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 1325 . . . . . . 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 7199 . . . . . . . . 9 |- ( q e. ( A X. B ) -> ( 2nd ` q ) e. B )
2423ad2antrl 764 . . . . . . . 8 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 2nd ` q ) e. B )
25 xp2nd 7199 . . . . . . . . 9 |- ( r e. ( A X. B ) -> ( 2nd ` r ) e. B )
2625ad2antll 765 . . . . . . . 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 4635 . . . . . . . . . . 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 208 . . . . . . . . . 10 |- ( ph -> A. b e. B A. d e. B ( b = d \/ ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) ) )
30 eqeq1 2626 . . . . . . . . . . . 12 |- ( b = ( 2nd ` q ) -> ( b = d <-> ( 2nd ` q ) = d ) )
31 csbeq1 3536 . . . . . . . . . . . . . 14 |- ( b = ( 2nd ` q ) -> [_ b / y ]_ D = [_ ( 2nd ` q ) / y ]_ D )
3231ineq1d 3813 . . . . . . . . . . . . 13 |- ( b = ( 2nd ` q ) -> ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) )
3332eqeq1d 2624 . . . . . . . . . . . 12 |- ( b = ( 2nd ` q ) -> ( ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) <-> ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = (/) ) )
3430, 33orbi12d 746 . . . . . . . . . . 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 2633 . . . . . . . . . . . 12 |- ( d = ( 2nd ` r ) -> ( ( 2nd ` q ) = d <-> ( 2nd ` q ) = ( 2nd ` r ) ) )
36 csbeq1 3536 . . . . . . . . . . . . . 14 |- ( d = ( 2nd ` r ) -> [_ d / y ]_ D = [_ ( 2nd ` r ) / y ]_ D )
3736ineq2d 3814 . . . . . . . . . . . . 13 |- ( d = ( 2nd ` r ) -> ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) )
3837eqeq1d 2624 . . . . . . . . . . . 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 746 . . . . . . . . . . 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 3322 . . . . . . . . . 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 34 . . . . . . . . 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 445 . . . . . . . 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 1325 . . . . . . 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 554 . . . . . 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 914 . . . . . 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 208 . . . . 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 546 . . . . 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 208 . . . 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 7207 . . . . . . 7 |- ( ( q e. ( A X. B ) /\ r e. ( A X. B ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) <-> q = r ) )
5049adantl 482 . . . . . 6 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) <-> q = r ) )
5150biimpd 219 . . . . 5 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) -> q = r ) )
52 inss2 3834 . . . . . . . . . 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 4010 . . . . . . . . . . . 12 |- [_ q / p ]_ ( E i^i F ) = ( [_ q / p ]_ E i^i [_ q / p ]_ F )
54 csbin 4010 . . . . . . . . . . . 12 |- [_ r / p ]_ ( E i^i F ) = ( [_ r / p ]_ E i^i [_ r / p ]_ F )
5553, 54ineq12i 3812 . . . . . . . . . . 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 3829 . . . . . . . . . . 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 2644 . . . . . . . . . 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 3203 . . . . . . . . . . . . 13 |- q e. _V
59 csbnestg 3998 . . . . . . . . . . . . 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 6201 . . . . . . . . . . . . . 14 |- ( 2nd ` p ) e. _V
62 disjxpin.2 . . . . . . . . . . . . . 14 |- ( y = ( 2nd ` p ) -> D = F )
6361, 62csbie 3559 . . . . . . . . . . . . 13 |- [_ ( 2nd ` p ) / y ]_ D = F
6463csbeq2i 3993 . . . . . . . . . . . 12 |- [_ q / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ q / p ]_ F
65 csbfv 6233 . . . . . . . . . . . . 13 |- [_ q / p ]_ ( 2nd ` p ) = ( 2nd ` q )
66 csbeq1 3536 . . . . . . . . . . . . 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 2653 . . . . . . . . . . 11 |- [_ ( 2nd ` q ) / y ]_ D = [_ q / p ]_ F
69 vex 3203 . . . . . . . . . . . . 13 |- r e. _V
70 csbnestg 3998 . . . . . . . . . . . . 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 3993 . . . . . . . . . . . 12 |- [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ r / p ]_ F
73 csbfv 6233 . . . . . . . . . . . . 13 |- [_ r / p ]_ ( 2nd ` p ) = ( 2nd ` r )
74 csbeq1 3536 . . . . . . . . . . . . 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 2653 . . . . . . . . . . 11 |- [_ ( 2nd ` r ) / y ]_ D = [_ r / p ]_ F
7768, 76ineq12i 3812 . . . . . . . . . 10 |- ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = ( [_ q / p ]_ F i^i [_ r / p ]_ F )
7852, 57, 773sstr4i 3644 . . . . . . . . 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 3975 . . . . . . . . 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 706 . . . . . . . 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 483 . . . . . 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 3833 . . . . . . . . . . 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 3998 . . . . . . . . . . . . . 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 6201 . . . . . . . . . . . . . . 15 |- ( 1st ` p ) e. _V
87 disjxpin.1 . . . . . . . . . . . . . . 15 |- ( x = ( 1st ` p ) -> C = E )
8886, 87csbie 3559 . . . . . . . . . . . . . 14 |- [_ ( 1st ` p ) / x ]_ C = E
8988csbeq2i 3993 . . . . . . . . . . . . 13 |- [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ q / p ]_ E
90 csbfv 6233 . . . . . . . . . . . . . 14 |- [_ q / p ]_ ( 1st ` p ) = ( 1st ` q )
91 csbeq1 3536 . . . . . . . . . . . . . 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 2653 . . . . . . . . . . . 12 |- [_ ( 1st ` q ) / x ]_ C = [_ q / p ]_ E
94 csbnestg 3998 . . . . . . . . . . . . . 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 3993 . . . . . . . . . . . . 13 |- [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ r / p ]_ E
97 csbfv 6233 . . . . . . . . . . . . . 14 |- [_ r / p ]_ ( 1st ` p ) = ( 1st ` r )
98 csbeq1 3536 . . . . . . . . . . . . . 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 2653 . . . . . . . . . . . 12 |- [_ ( 1st ` r ) / x ]_ C = [_ r / p ]_ E
10193, 100ineq12i 3812 . . . . . . . . . . 11 |- ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = ( [_ q / p ]_ E i^i [_ r / p ]_ E )
10283, 57, 1013sstr4i 3644 . . . . . . . . . 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 3975 . . . . . . . . . 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 706 . . . . . . . . 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 484 . . . . . . 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 483 . . . . . . 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 395 . . . . . 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 395 . . . . 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 883 . . . 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 2971 . 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 4635 . 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 224 1 |- ( ph -> Disj p e. ( A X. B ) ( E i^i F ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [_csb 3533   i^i cin 3573   C_ wss 3574   (/)c0 3915  Disj wdisj 4620   X. cxp 5112   ` cfv 5888   1stc1st 7166   2ndc2nd 7167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169
This theorem is referenced by:  sibfof  30402
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