Theorem disjprg 4648

Description: A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
disjprg.1	|- ( x = A -> C = D )
disjprg.2	|- ( x = B -> C = E )

Assertion

Ref	Expression
disjprg	|- ( ( A e. V /\ B e. V /\ A =/= B ) -> (Disj x e. { A , B } C <-> ( D i^i E ) = (/) ) )

Distinct variable groups:   x, A   x, B   x, D   x, E

Allowed substitution hints:   C( x)   V( x)

Proof of Theorem disjprg

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . . . . 7 |- ( y = A -> ( y = z <-> A = z ) )
2		nfcv 2764	. . . . . . . . . 10 |- F/_ x A
3		nfcv 2764	. . . . . . . . . 10 |- F/_ x D
4		disjprg.1	. . . . . . . . . 10 |- ( x = A -> C = D )
5	2, 3, 4	csbhypf 3552	. . . . . . . . 9 |- ( y = A -> [_ y / x ]_ C = D )
6	5	ineq1d 3813	. . . . . . . 8 |- ( y = A -> ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = ( D i^i [_ z / x ]_ C ) )
7	6	eqeq1d 2624	. . . . . . 7 |- ( y = A -> ( ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) <-> ( D i^i [_ z / x ]_ C ) = (/) ) )
8	1, 7	orbi12d 746	. . . . . 6 |- ( y = A -> ( ( y = z \/ ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) ) <-> ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) ) )
9	8	ralbidv 2986	. . . . 5 |- ( y = A -> ( A. z e. { A , B } ( y = z \/ ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) ) <-> A. z e. { A , B } ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) ) )
10		eqeq1 2626	. . . . . . 7 |- ( y = B -> ( y = z <-> B = z ) )
11		nfcv 2764	. . . . . . . . . 10 |- F/_ x B
12		nfcv 2764	. . . . . . . . . 10 |- F/_ x E
13		disjprg.2	. . . . . . . . . 10 |- ( x = B -> C = E )
14	11, 12, 13	csbhypf 3552	. . . . . . . . 9 |- ( y = B -> [_ y / x ]_ C = E )
15	14	ineq1d 3813	. . . . . . . 8 |- ( y = B -> ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = ( E i^i [_ z / x ]_ C ) )
16	15	eqeq1d 2624	. . . . . . 7 |- ( y = B -> ( ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) <-> ( E i^i [_ z / x ]_ C ) = (/) ) )
17	10, 16	orbi12d 746	. . . . . 6 |- ( y = B -> ( ( y = z \/ ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) ) <-> ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) ) )
18	17	ralbidv 2986	. . . . 5 |- ( y = B -> ( A. z e. { A , B } ( y = z \/ ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) ) <-> A. z e. { A , B } ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) ) )
19	9, 18	ralprg 4234	. . . 4 |- ( ( A e. V /\ B e. V ) -> ( A. y e. { A , B } A. z e. { A , B } ( y = z \/ ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) ) <-> ( A. z e. { A , B } ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) /\ A. z e. { A , B } ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) ) ) )
20	19	3adant3 1081	. . 3 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( A. y e. { A , B } A. z e. { A , B } ( y = z \/ ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) ) <-> ( A. z e. { A , B } ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) /\ A. z e. { A , B } ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) ) ) )
21		id 22	. . . . . . . . . 10 |- ( z = A -> z = A )
22	21	eqcomd 2628	. . . . . . . . 9 |- ( z = A -> A = z )
23	22	orcd 407	. . . . . . . 8 |- ( z = A -> ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) )
24		a1tru 1500	. . . . . . . 8 |- ( z = A -> T. )
25	23, 24	2thd 255	. . . . . . 7 |- ( z = A -> ( ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) <-> T. ) )
26		eqeq2 2633	. . . . . . . 8 |- ( z = B -> ( A = z <-> A = B ) )
27	11, 12, 13	csbhypf 3552	. . . . . . . . . 10 |- ( z = B -> [_ z / x ]_ C = E )
28	27	ineq2d 3814	. . . . . . . . 9 |- ( z = B -> ( D i^i [_ z / x ]_ C ) = ( D i^i E ) )
29	28	eqeq1d 2624	. . . . . . . 8 |- ( z = B -> ( ( D i^i [_ z / x ]_ C ) = (/) <-> ( D i^i E ) = (/) ) )
30	26, 29	orbi12d 746	. . . . . . 7 |- ( z = B -> ( ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) <-> ( A = B \/ ( D i^i E ) = (/) ) ) )
31	25, 30	ralprg 4234	. . . . . 6 |- ( ( A e. V /\ B e. V ) -> ( A. z e. { A , B } ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) <-> ( T. /\ ( A = B \/ ( D i^i E ) = (/) ) ) ) )
32	31	3adant3 1081	. . . . 5 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( A. z e. { A , B } ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) <-> ( T. /\ ( A = B \/ ( D i^i E ) = (/) ) ) ) )
33		simp3 1063	. . . . . . . 8 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> A =/= B )
34	33	neneqd 2799	. . . . . . 7 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> -. A = B )
35		biorf 420	. . . . . . 7 |- ( -. A = B -> ( ( D i^i E ) = (/) <-> ( A = B \/ ( D i^i E ) = (/) ) ) )
36	34, 35	syl 17	. . . . . 6 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( ( D i^i E ) = (/) <-> ( A = B \/ ( D i^i E ) = (/) ) ) )
37		tru 1487	. . . . . . 7 |- T.
38	37	biantrur 527	. . . . . 6 |- ( ( A = B \/ ( D i^i E ) = (/) ) <-> ( T. /\ ( A = B \/ ( D i^i E ) = (/) ) ) )
39	36, 38	syl6bb 276	. . . . 5 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( ( D i^i E ) = (/) <-> ( T. /\ ( A = B \/ ( D i^i E ) = (/) ) ) ) )
40	32, 39	bitr4d 271	. . . 4 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( A. z e. { A , B } ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) <-> ( D i^i E ) = (/) ) )
41		eqeq2 2633	. . . . . . . . 9 |- ( z = A -> ( B = z <-> B = A ) )
42		eqcom 2629	. . . . . . . . 9 |- ( B = A <-> A = B )
43	41, 42	syl6bb 276	. . . . . . . 8 |- ( z = A -> ( B = z <-> A = B ) )
44	2, 3, 4	csbhypf 3552	. . . . . . . . . . 11 |- ( z = A -> [_ z / x ]_ C = D )
45	44	ineq2d 3814	. . . . . . . . . 10 |- ( z = A -> ( E i^i [_ z / x ]_ C ) = ( E i^i D ) )
46		incom 3805	. . . . . . . . . 10 |- ( E i^i D ) = ( D i^i E )
47	45, 46	syl6eq 2672	. . . . . . . . 9 |- ( z = A -> ( E i^i [_ z / x ]_ C ) = ( D i^i E ) )
48	47	eqeq1d 2624	. . . . . . . 8 |- ( z = A -> ( ( E i^i [_ z / x ]_ C ) = (/) <-> ( D i^i E ) = (/) ) )
49	43, 48	orbi12d 746	. . . . . . 7 |- ( z = A -> ( ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) <-> ( A = B \/ ( D i^i E ) = (/) ) ) )
50		id 22	. . . . . . . . . 10 |- ( z = B -> z = B )
51	50	eqcomd 2628	. . . . . . . . 9 |- ( z = B -> B = z )
52	51	orcd 407	. . . . . . . 8 |- ( z = B -> ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) )
53		a1tru 1500	. . . . . . . 8 |- ( z = B -> T. )
54	52, 53	2thd 255	. . . . . . 7 |- ( z = B -> ( ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) <-> T. ) )
55	49, 54	ralprg 4234	. . . . . 6 |- ( ( A e. V /\ B e. V ) -> ( A. z e. { A , B } ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) <-> ( ( A = B \/ ( D i^i E ) = (/) ) /\ T. ) ) )
56	55	3adant3 1081	. . . . 5 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( A. z e. { A , B } ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) <-> ( ( A = B \/ ( D i^i E ) = (/) ) /\ T. ) ) )
57	37	biantru 526	. . . . . 6 |- ( ( A = B \/ ( D i^i E ) = (/) ) <-> ( ( A = B \/ ( D i^i E ) = (/) ) /\ T. ) )
58	36, 57	syl6bb 276	. . . . 5 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( ( D i^i E ) = (/) <-> ( ( A = B \/ ( D i^i E ) = (/) ) /\ T. ) ) )
59	56, 58	bitr4d 271	. . . 4 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( A. z e. { A , B } ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) <-> ( D i^i E ) = (/) ) )
60	40, 59	anbi12d 747	. . 3 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( ( A. z e. { A , B } ( A = z \/ ( D i^i [_ z / x ]_ C ) = (/) ) /\ A. z e. { A , B } ( B = z \/ ( E i^i [_ z / x ]_ C ) = (/) ) ) <-> ( ( D i^i E ) = (/) /\ ( D i^i E ) = (/) ) ) )
61	20, 60	bitrd 268	. 2 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> ( A. y e. { A , B } A. z e. { A , B } ( y = z \/ ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) ) <-> ( ( D i^i E ) = (/) /\ ( D i^i E ) = (/) ) ) )
62		disjors 4635	. 2 |- (Disj x e. { A , B } C <-> A. y e. { A , B } A. z e. { A , B } ( y = z \/ ( [_ y / x ]_ C i^i [_ z / x ]_ C ) = (/) ) )
63		pm4.24 675	. 2 |- ( ( D i^i E ) = (/) <-> ( ( D i^i E ) = (/) /\ ( D i^i E ) = (/) ) )
64	61, 62, 63	3bitr4g 303	1 |- ( ( A e. V /\ B e. V /\ A =/= B ) -> (Disj x e. { A , B } C <-> ( D i^i E ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   i^i cin 3573   (/)c0 3915   {cpr 4179  Disj wdisj 4620

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-pr 4180  df-disj 4621

This theorem is referenced by:  disjdifprg  29388  unelldsys  30221  pmeasmono  30386  probun  30481  meadjun  40679