Theorem disjxpin 27148

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 6814	. . . . . . . . 9 |- ( q e. ( A X. B ) -> ( 1st ` q ) e. A )
2	1	ad2antrl 727	. . . . . . . 8 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 1st ` q ) e. A )
3		xp1st 6814	. . . . . . . . 9 |- ( r e. ( A X. B ) -> ( 1st ` r ) e. A )
4	3	ad2antll 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 4433	. . . . . . . . . . 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 196	. . . . . . . . . 10 |- ( ph -> A. a e. A A. c e. A ( a = c \/ ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = (/) ) )
9		eqeq1 2471	. . . . . . . . . . . 12 |- ( a = ( 1st ` q ) -> ( a = c <-> ( 1st ` q ) = c ) )
10		csbeq1 3438	. . . . . . . . . . . . . 14 |- ( a = ( 1st ` q ) -> [_ a / x ]_ C = [_ ( 1st ` q ) / x ]_ C )
11	10	ineq1d 3699	. . . . . . . . . . . . 13 |- ( a = ( 1st ` q ) -> ( [_ a / x ]_ C i^i [_ c / x ]_ C ) = ( [_ ( 1st ` q ) / x ]_ C i^i [_ c / x ]_ C ) )
12	11	eqeq1d 2469	. . . . . . . . . . . 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 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 2482	. . . . . . . . . . . 12 |- ( c = ( 1st ` r ) -> ( ( 1st ` q ) = c <-> ( 1st ` q ) = ( 1st ` r ) ) )
15		csbeq1 3438	. . . . . . . . . . . . . 14 |- ( c = ( 1st ` r ) -> [_ c / x ]_ C = [_ ( 1st ` r ) / x ]_ C )
16	15	ineq2d 3700	. . . . . . . . . . . . 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 2469	. . . . . . . . . . . 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 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 ) = (/) ) ) )
19	13, 18	rspc2v 3223	. . . . . . . . . 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 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 ) = (/) ) ) )
21	20	imp 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 ) = (/) ) )
22	2, 4, 5, 21	syl21anc 1227	. . . . . . 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 6815	. . . . . . . . 9 |- ( q e. ( A X. B ) -> ( 2nd ` q ) e. B )
24	23	ad2antrl 727	. . . . . . . 8 |- ( ( ph /\ ( q e. ( A X. B ) /\ r e. ( A X. B ) ) ) -> ( 2nd ` q ) e. B )
25		xp2nd 6815	. . . . . . . . 9 |- ( r e. ( A X. B ) -> ( 2nd ` r ) e. B )
26	25	ad2antll 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 4433	. . . . . . . . . . 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 196	. . . . . . . . . 10 |- ( ph -> A. b e. B A. d e. B ( b = d \/ ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = (/) ) )
30		eqeq1 2471	. . . . . . . . . . . 12 |- ( b = ( 2nd ` q ) -> ( b = d <-> ( 2nd ` q ) = d ) )
31		csbeq1 3438	. . . . . . . . . . . . . 14 |- ( b = ( 2nd ` q ) -> [_ b / y ]_ D = [_ ( 2nd ` q ) / y ]_ D )
32	31	ineq1d 3699	. . . . . . . . . . . . 13 |- ( b = ( 2nd ` q ) -> ( [_ b / y ]_ D i^i [_ d / y ]_ D ) = ( [_ ( 2nd ` q ) / y ]_ D i^i [_ d / y ]_ D ) )
33	32	eqeq1d 2469	. . . . . . . . . . . 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 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 2482	. . . . . . . . . . . 12 |- ( d = ( 2nd ` r ) -> ( ( 2nd ` q ) = d <-> ( 2nd ` q ) = ( 2nd ` r ) ) )
36		csbeq1 3438	. . . . . . . . . . . . . 14 |- ( d = ( 2nd ` r ) -> [_ d / y ]_ D = [_ ( 2nd ` r ) / y ]_ D )
37	36	ineq2d 3700	. . . . . . . . . . . . 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 2469	. . . . . . . . . . . 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 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 ) = (/) ) ) )
40	34, 39	rspc2v 3223	. . . . . . . . . 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 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 ) = (/) ) ) )
42	41	imp 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 ) = (/) ) )
43	24, 26, 5, 42	syl21anc 1227	. . . . . . 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 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 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 ) = (/) ) ) ) )
46	44, 45	sylib 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 ) = (/) ) ) ) ) )
48	46, 47	sylib 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 6823	. . . . . . 7 |- ( ( q e. ( A X. B ) /\ r e. ( A X. B ) ) -> ( ( ( 1st ` q ) = ( 1st ` r ) /\ ( 2nd ` q ) = ( 2nd ` r ) ) <-> q = r ) )
50	49	adantl 466	. . . . . 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 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 3719	. . . . . . . . . 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 3860	. . . . . . . . . . . 12 |- [_ q / p ]_ ( E i^i F ) = ( [_ q / p ]_ E i^i [_ q / p ]_ F )
54		csbin 3860	. . . . . . . . . . . 12 |- [_ r / p ]_ ( E i^i F ) = ( [_ r / p ]_ E i^i [_ r / p ]_ F )
55	53, 54	ineq12i 3698	. . . . . . . . . . 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 3714	. . . . . . . . . . 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 2496	. . . . . . . . . 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 3116	. . . . . . . . . . . . 13 |- q e. _V
59		csbnestg 3842	. . . . . . . . . . . . 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 5876	. . . . . . . . . . . . . 14 |- ( 2nd ` p ) e. _V
62		nfcv 2629	. . . . . . . . . . . . . 14 |- F/_ y F
63		disjxpin.2	. . . . . . . . . . . . . 14 |- ( y = ( 2nd ` p ) -> D = F )
64	61, 62, 63	csbief 3460	. . . . . . . . . . . . 13 |- [_ ( 2nd ` p ) / y ]_ D = F
65	64	csbeq2i 3836	. . . . . . . . . . . 12 |- [_ q / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ q / p ]_ F
66		nfcv 2629	. . . . . . . . . . . . . 14 |- F/_ p ( 2nd ` q )
67		fveq2 5866	. . . . . . . . . . . . . 14 |- ( p = q -> ( 2nd ` p ) = ( 2nd ` q ) )
68	58, 66, 67	csbief 3460	. . . . . . . . . . . . 13 |- [_ q / p ]_ ( 2nd ` p ) = ( 2nd ` q )
69		csbeq1 3438	. . . . . . . . . . . . 13 |- ( [_ q / p ]_ ( 2nd ` p ) = ( 2nd ` q ) -> [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` q ) / y ]_ D )
70	68, 69	ax-mp 5	. . . . . . . . . . . 12 |- [_ [_ q / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` q ) / y ]_ D
71	60, 65, 70	3eqtr3ri 2505	. . . . . . . . . . 11 |- [_ ( 2nd ` q ) / y ]_ D = [_ q / p ]_ F
72		vex 3116	. . . . . . . . . . . . 13 |- r e. _V
73		csbnestg 3842	. . . . . . . . . . . . 13 |- ( r e. _V -> [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D )
74	72, 73	ax-mp 5	. . . . . . . . . . . 12 |- [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D
75	64	csbeq2i 3836	. . . . . . . . . . . 12 |- [_ r / p ]_ [_ ( 2nd ` p ) / y ]_ D = [_ r / p ]_ F
76		nfcv 2629	. . . . . . . . . . . . . 14 |- F/_ p ( 2nd ` r )
77		fveq2 5866	. . . . . . . . . . . . . 14 |- ( p = r -> ( 2nd ` p ) = ( 2nd ` r ) )
78	72, 76, 77	csbief 3460	. . . . . . . . . . . . 13 |- [_ r / p ]_ ( 2nd ` p ) = ( 2nd ` r )
79		csbeq1 3438	. . . . . . . . . . . . 13 |- ( [_ r / p ]_ ( 2nd ` p ) = ( 2nd ` r ) -> [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` r ) / y ]_ D )
80	78, 79	ax-mp 5	. . . . . . . . . . . 12 |- [_ [_ r / p ]_ ( 2nd ` p ) / y ]_ D = [_ ( 2nd ` r ) / y ]_ D
81	74, 75, 80	3eqtr3ri 2505	. . . . . . . . . . 11 |- [_ ( 2nd ` r ) / y ]_ D = [_ r / p ]_ F
82	71, 81	ineq12i 3698	. . . . . . . . . 10 |- ( [_ ( 2nd ` q ) / y ]_ D i^i [_ ( 2nd ` r ) / y ]_ D ) = ( [_ q / p ]_ F i^i [_ r / p ]_ F )
83	52, 57, 82	3sstr4i 3543	. . . . . . . . 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 3817	. . . . . . . . 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 ) ) = (/) )
85	83, 84	mpan 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 ) ) = (/) )
86	85	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 ) ) = (/) ) )
87	86	adantld 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 3718	. . . . . . . . . . 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 3842	. . . . . . . . . . . . . 14 |- ( q e. _V -> [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C )
90	58, 89	ax-mp 5	. . . . . . . . . . . . 13 |- [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C
91		fvex 5876	. . . . . . . . . . . . . . 15 |- ( 1st ` p ) e. _V
92		nfcv 2629	. . . . . . . . . . . . . . 15 |- F/_ x E
93		disjxpin.1	. . . . . . . . . . . . . . 15 |- ( x = ( 1st ` p ) -> C = E )
94	91, 92, 93	csbief 3460	. . . . . . . . . . . . . 14 |- [_ ( 1st ` p ) / x ]_ C = E
95	94	csbeq2i 3836	. . . . . . . . . . . . 13 |- [_ q / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ q / p ]_ E
96		nfcv 2629	. . . . . . . . . . . . . . 15 |- F/_ p ( 1st ` q )
97		fveq2 5866	. . . . . . . . . . . . . . 15 |- ( p = q -> ( 1st ` p ) = ( 1st ` q ) )
98	58, 96, 97	csbief 3460	. . . . . . . . . . . . . 14 |- [_ q / p ]_ ( 1st ` p ) = ( 1st ` q )
99		csbeq1 3438	. . . . . . . . . . . . . 14 |- ( [_ q / p ]_ ( 1st ` p ) = ( 1st ` q ) -> [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` q ) / x ]_ C )
100	98, 99	ax-mp 5	. . . . . . . . . . . . 13 |- [_ [_ q / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` q ) / x ]_ C
101	90, 95, 100	3eqtr3ri 2505	. . . . . . . . . . . 12 |- [_ ( 1st ` q ) / x ]_ C = [_ q / p ]_ E
102		csbnestg 3842	. . . . . . . . . . . . . 14 |- ( r e. _V -> [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C )
103	72, 102	ax-mp 5	. . . . . . . . . . . . 13 |- [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C
104	94	csbeq2i 3836	. . . . . . . . . . . . 13 |- [_ r / p ]_ [_ ( 1st ` p ) / x ]_ C = [_ r / p ]_ E
105		nfcv 2629	. . . . . . . . . . . . . . 15 |- F/_ p ( 1st ` r )
106		fveq2 5866	. . . . . . . . . . . . . . 15 |- ( p = r -> ( 1st ` p ) = ( 1st ` r ) )
107	72, 105, 106	csbief 3460	. . . . . . . . . . . . . 14 |- [_ r / p ]_ ( 1st ` p ) = ( 1st ` r )
108		csbeq1 3438	. . . . . . . . . . . . . 14 |- ( [_ r / p ]_ ( 1st ` p ) = ( 1st ` r ) -> [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` r ) / x ]_ C )
109	107, 108	ax-mp 5	. . . . . . . . . . . . 13 |- [_ [_ r / p ]_ ( 1st ` p ) / x ]_ C = [_ ( 1st ` r ) / x ]_ C
110	103, 104, 109	3eqtr3ri 2505	. . . . . . . . . . . 12 |- [_ ( 1st ` r ) / x ]_ C = [_ r / p ]_ E
111	101, 110	ineq12i 3698	. . . . . . . . . . 11 |- ( [_ ( 1st ` q ) / x ]_ C i^i [_ ( 1st ` r ) / x ]_ C ) = ( [_ q / p ]_ E i^i [_ r / p ]_ E )
112	88, 57, 111	3sstr4i 3543	. . . . . . . . . 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 3817	. . . . . . . . . 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 ) ) = (/) )
114	112, 113	mpan 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 ) ) = (/) )
115	114	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 ) ) = (/) ) )
116	115	adantrd 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 ) ) = (/) ) )
117	86	adantld 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 ) ) = (/) ) )
118	116, 117	jaod 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 ) ) = (/) ) )
119	87, 118	jaod 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 ) ) = (/) ) )
120	51, 119	orim12d 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 ) ) = (/) ) ) )
121	48, 120	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 ) ) = (/) ) )
122	121	ralrimivva 2885	. 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 4433	. 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 ) ) = (/) ) )
124	122, 123	sylibr 212	1 |- ( ph -> Disj p e. ( A X. B ) ( E i^i F ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   _Vcvv 3113   [_csb 3435   i^i cin 3475   C_ wss 3476   (/)c0 3785  Disj wdisj 4417   X. cxp 4997   ` cfv 5588   1stc1st 6782   2ndc2nd 6783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-1st 6784  df-2nd 6785

This theorem is referenced by:  sibfof  27950