Theorem disjxpin 28198

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.

Step	Hyp	Ref	Expression
1		xp1st 6823	. . . . . . . . 9 |- ( q e. ( A X. B ) -> ( 1st ` q ) e. A )
2	1	ad2antrl 734	. . . . . . . 8 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 1st ` q ) e. A )
3		xp1st 6823	. . . . . . . . 9 |- ( r e. ( A X. B ) -> ( 1st ` r ) e. A )
4	3	ad2antll 735	. . . . . . . 8 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 1st ` r ) e. A )
5		simpl 459	. . . . . . . 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 4388	. . . . . . . . . . 11 |- (Disj x e. A C <-> A. a e. A A. c e. A ( a = c \/ ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) ) )
8	6, 7	sylib 200	. . . . . . . . . 10 |- ( ph -> A. a e. A A. c e. A ( a = c \/ ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) ) )
9		eqeq1 2455	. . . . . . . . . . . 12 |- ( a = ( 1st ` q ) -> ( a = c <-> ( 1st ` q ) = c ) )
10		csbeq1 3366	. . . . . . . . . . . . . 14 |- ( a = ( 1st ` q ) -> [_ a / x ]_ C = [_ ( 1st ` q ) / x ]_ C )
11	10	ineq1d 3633	. . . . . . . . . . . . 13 |- ( a = ( 1st ` q ) -> ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) )
12	11	eqeq1d 2453	. . . . . . . . . . . 12 |- ( a = ( 1st ` q ) -> ( ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) <-> ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = (/) ) )
13	9, 12	orbi12d 716	. . . . . . . . . . 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 2462	. . . . . . . . . . . 12 |- ( c = ( 1st ` r ) -> ( ( 1st ` q ) = c <-> ( 1st ` q ) = ( 1st ` r ) ) )
15		csbeq1 3366	. . . . . . . . . . . . . 14 |- ( c = ( 1st ` r ) -> [_ c / x ]_ C = [_ ( 1st ` r ) / x ]_ C )
16	15	ineq2d 3634	. . . . . . . . . . . . 13 |- ( c = ( 1st ` r ) -> ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) )
17	16	eqeq1d 2453	. . . . . . . . . . . 12 |- ( c = ( 1st ` r ) -> ( ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) = (/) <-> ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) )
18	14, 17	orbi12d 716	. . . . . . . . . . 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 ) = (/) ) ) )
19	13, 18	rspc2v 3159	. . . . . . . . . 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 ) = (/) ) ) )
20	8, 19	syl5 33	. . . . . . . . 9 |- ( ( ( 1st ` q ) e. A /\ ( 1st ` r ) e. A ) -> ( ph -> ( ( 1st ` q ) = ( 1st ` r ) \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) ) )
21	20	imp 431	. . . . . . . 8 |- ( ( ( ( 1st ` q ) e. A /\ ( 1st ` r ) e. A ) /\ ph ) -> ( ( 1st ` q ) = ( 1st ` r ) \/ ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = (/) ) )
22	2, 4, 5, 21	syl21anc 1267	. . . . . . 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 6824	. . . . . . . . 9 |- ( q e. ( A X. B ) -> ( 2nd ` q ) e. B )
24	23	ad2antrl 734	. . . . . . . 8 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 2nd ` q ) e. B )
25		xp2nd 6824	. . . . . . . . 9 |- ( r e. ( A X. B ) -> ( 2nd ` r ) e. B )
26	25	ad2antll 735	. . . . . . . 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 4388	. . . . . . . . . . 11 |- (Disj y e. B D <-> A. b e. B A. d e. B ( b = d \/ ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) ) )
29	27, 28	sylib 200	. . . . . . . . . 10 |- ( ph -> A. b e. B A. d e. B ( b = d \/ ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) ) )
30		eqeq1 2455	. . . . . . . . . . . 12 |- ( b = ( 2nd ` q ) -> ( b = d <-> ( 2nd ` q ) = d ) )
31		csbeq1 3366	. . . . . . . . . . . . . 14 |- ( b = ( 2nd ` q ) -> [_ b / y ]_ D = [_ ( 2nd ` q ) / y ]_ D )
32	31	ineq1d 3633	. . . . . . . . . . . . 13 |- ( b = ( 2nd ` q ) -> ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) )
33	32	eqeq1d 2453	. . . . . . . . . . . 12 |- ( b = ( 2nd ` q ) -> ( ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) <-> ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = (/) ) )
34	30, 33	orbi12d 716	. . . . . . . . . . 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 2462	. . . . . . . . . . . 12 |- ( d = ( 2nd ` r ) -> ( ( 2nd ` q ) = d <-> ( 2nd ` q ) = ( 2nd ` r ) ) )
36		csbeq1 3366	. . . . . . . . . . . . . 14 |- ( d = ( 2nd ` r ) -> [_ d / y ]_ D = [_ ( 2nd ` r ) / y ]_ D )
37	36	ineq2d 3634	. . . . . . . . . . . . 13 |- ( d = ( 2nd ` r ) -> ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) )
38	37	eqeq1d 2453	. . . . . . . . . . . 12 |- ( d = ( 2nd ` r ) -> ( ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) = (/) <-> ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) )
39	35, 38	orbi12d 716	. . . . . . . . . . 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 ) = (/) ) ) )
40	34, 39	rspc2v 3159	. . . . . . . . . 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 ) = (/) ) ) )
41	29, 40	syl5 33	. . . . . . . . 9 |- ( ( ( 2nd ` q ) e. B /\ ( 2nd ` r ) e. B ) -> ( ph -> ( ( 2nd ` q ) = ( 2nd ` r ) \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) ) )
42	41	imp 431	. . . . . . . 8 |- ( ( ( ( 2nd ` q ) e. B /\ ( 2nd ` r ) e. B ) /\ ph ) -> ( ( 2nd ` q ) = ( 2nd ` r ) \/ ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = (/) ) )
43	24, 26, 5, 42	syl21anc 1267	. . . . . . 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 ) = (/) ) )
44	22, 43	jca 535	. . . . . 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 881	. . . . . 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 ) = (/) ) ) ) )
46	44, 45	sylib 200	. . . . 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 527	. . . . 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 ) = (/) ) ) ) ) )
48	46, 47	sylib 200	. . . 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 6832	. . . . . . 7 |- ( ( q e. ( A X. B ) /\ r e. ( A X. B ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) <-> q = r ) )
50	49	adantl 468	. . . . . 6 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) <-> q = r ) )
51	50	biimpd 211	. . . . 5 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) -> q = r ) )
52		inss2 3653	. . . . . . . . . 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 3799	. . . . . . . . . . . 12 |- [_ q / p ]_ ( E i^i F ) = ( [_ q / p ]_ E i^i [_ q / p ]_ F )
54		csbin 3799	. . . . . . . . . . . 12 |- [_ r / p ]_ ( E i^i F ) = ( [_ r / p ]_ E i^i [_ r / p ]_ F )
55	53, 54	ineq12i 3632	. . . . . . . . . . 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 3648	. . . . . . . . . . 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 ) )
57	55, 56	eqtri 2473	. . . . . . . . . 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 3048	. . . . . . . . . . . . 13 |- q e. _V
59		csbnestg 3787	. . . . . . . . . . . . 13 |- ( q e. _V -> [_ q / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D )
60	58, 59	ax-mp 5	. . . . . . . . . . . 12 |- [_ q / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D
61		fvex 5875	. . . . . . . . . . . . . 14 |- ( 2nd ` p ) e. _V
62		disjxpin.2	. . . . . . . . . . . . . 14 |- ( y = ( 2nd ` p ) -> D = F )
63	61, 62	csbie 3389	. . . . . . . . . . . . 13 |- [_ ( 2nd ` p ) / y ]_ D = F
64	63	csbeq2i 3782	. . . . . . . . . . . 12 |- [_ q / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ q / p ]_ F
65		csbfv 5902	. . . . . . . . . . . . 13 |- [_ q / p ]_ ( 2nd ` p ) = ( 2nd ` q )
66		csbeq1 3366	. . . . . . . . . . . . 13 |- ( [_ q / p ]_ ( 2nd ` p ) = ( 2nd ` q ) -> [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` q ) / y ]_ D )
67	65, 66	ax-mp 5	. . . . . . . . . . . 12 |- [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` q ) / y ]_ D
68	60, 64, 67	3eqtr3ri 2482	. . . . . . . . . . 11 |- [_ ( 2nd ` q ) / y ]_ D = [_ q / p ]_ F
69		vex 3048	. . . . . . . . . . . . 13 |- r e. _V
70		csbnestg 3787	. . . . . . . . . . . . 13 |- ( r e. _V -> [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D )
71	69, 70	ax-mp 5	. . . . . . . . . . . 12 |- [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D
72	63	csbeq2i 3782	. . . . . . . . . . . 12 |- [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ r / p ]_ F
73		csbfv 5902	. . . . . . . . . . . . 13 |- [_ r / p ]_ ( 2nd ` p ) = ( 2nd ` r )
74		csbeq1 3366	. . . . . . . . . . . . 13 |- ( [_ r / p ]_ ( 2nd ` p ) = ( 2nd ` r ) -> [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` r ) / y ]_ D )
75	73, 74	ax-mp 5	. . . . . . . . . . . 12 |- [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` r ) / y ]_ D
76	71, 72, 75	3eqtr3ri 2482	. . . . . . . . . . 11 |- [_ ( 2nd ` r ) / y ]_ D = [_ r / p ]_ F
77	68, 76	ineq12i 3632	. . . . . . . . . 10 |- ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = ( [_ q / p ]_ F i^i [_ r / p ]_ F )
78	52, 57, 77	3sstr4i 3471	. . . . . . . . 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 3766	. . . . . . . . 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 ) ) = (/) )
80	78, 79	mpan 676	. . . . . . . 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 ) ) = (/) )
81	80	a1i 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 ) ) = (/) ) )
82	81	adantld 469	. . . . . 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 3652	. . . . . . . . . . 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 3787	. . . . . . . . . . . . . 14 |- ( q e. _V -> [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C )
85	58, 84	ax-mp 5	. . . . . . . . . . . . 13 |- [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C
86		fvex 5875	. . . . . . . . . . . . . . 15 |- ( 1st ` p ) e. _V
87		disjxpin.1	. . . . . . . . . . . . . . 15 |- ( x = ( 1st ` p ) -> C = E )
88	86, 87	csbie 3389	. . . . . . . . . . . . . 14 |- [_ ( 1st ` p ) / x ]_ C = E
89	88	csbeq2i 3782	. . . . . . . . . . . . 13 |- [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ q / p ]_ E
90		csbfv 5902	. . . . . . . . . . . . . 14 |- [_ q / p ]_ ( 1st ` p ) = ( 1st ` q )
91		csbeq1 3366	. . . . . . . . . . . . . 14 |- ( [_ q / p ]_ ( 1st ` p ) = ( 1st ` q ) -> [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` q ) / x ]_ C )
92	90, 91	ax-mp 5	. . . . . . . . . . . . 13 |- [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` q ) / x ]_ C
93	85, 89, 92	3eqtr3ri 2482	. . . . . . . . . . . 12 |- [_ ( 1st ` q ) / x ]_ C = [_ q / p ]_ E
94		csbnestg 3787	. . . . . . . . . . . . . 14 |- ( r e. _V -> [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C )
95	69, 94	ax-mp 5	. . . . . . . . . . . . 13 |- [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C
96	88	csbeq2i 3782	. . . . . . . . . . . . 13 |- [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ r / p ]_ E
97		csbfv 5902	. . . . . . . . . . . . . 14 |- [_ r / p ]_ ( 1st ` p ) = ( 1st ` r )
98		csbeq1 3366	. . . . . . . . . . . . . 14 |- ( [_ r / p ]_ ( 1st ` p ) = ( 1st ` r ) -> [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` r ) / x ]_ C )
99	97, 98	ax-mp 5	. . . . . . . . . . . . 13 |- [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` r ) / x ]_ C
100	95, 96, 99	3eqtr3ri 2482	. . . . . . . . . . . 12 |- [_ ( 1st ` r ) / x ]_ C = [_ r / p ]_ E
101	93, 100	ineq12i 3632	. . . . . . . . . . 11 |- ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = ( [_ q / p ]_ E i^i [_ r / p ]_ E )
102	83, 57, 101	3sstr4i 3471	. . . . . . . . . 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 3766	. . . . . . . . . 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 ) ) = (/) )
104	102, 103	mpan 676	. . . . . . . . 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 ) ) = (/) )
105	104	a1i 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 ) ) = (/) ) )
106	105	adantrd 470	. . . . . . 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 ) ) = (/) ) )
107	81	adantld 469	. . . . . . 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 ) ) = (/) ) )
108	106, 107	jaod 382	. . . . . 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 ) ) = (/) ) )
109	82, 108	jaod 382	. . . . 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 ) ) = (/) ) )
110	51, 109	orim12d 849	. . . 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 ) ) = (/) ) ) )
111	48, 110	mpd 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 ) ) = (/) ) )
112	111	ralrimivva 2809	. 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 4388	. 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 ) ) = (/) ) )
114	112, 113	sylibr 216	1 |- ( ph -> Disj p e. ( A X. B ) ( E i^i F ) )

Syntax hints:   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   _Vcvv 3045   [_csb 3363   i^i cin 3403   C_ wss 3404   (/)c0 3731  Disj wdisj 4373   X. cxp 4832   ` cfv 5582   1stc1st 6791   2ndc2nd 6792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fv 5590  df-1st 6793  df-2nd 6794

This theorem is referenced by:  sibfof  29173