Theorem disjiund 4643

Description: Conditions for a collection of index unions of sets A ( a , b ) for a e. V and b e. W to be disjoint. (Contributed by AV, 9-Jan-2022.)

Hypotheses

Ref	Expression
disjiund.1	|- ( a = c -> A = C )
disjiund.2	|- ( b = d -> C = D )
disjiund.3	|- ( a = c -> W = X )
disjiund.4	|- ( ( ph /\ x e. A /\ x e. D ) -> a = c )

Assertion

Ref	Expression
disjiund	|- ( ph -> Disj a e. V U_ b e. W A )

Distinct variable groups:   A, c, d, x   C, a, d, x   D, b   V, a, c   W, b, c, d, x   X, a, b, d, x   ph, a, b, c, d, x

Allowed substitution hints:   A( a, b)   C( b, c)   D( x, a, c, d)   V( x, b, d)   W( a)   X( c)

Proof of Theorem disjiund

Step	Hyp	Ref	Expression
1		eliun 4524	. . . . . . . . 9 |- ( x e. U_ b e. W A <-> E. b e. W x e. A )
2		eliun 4524	. . . . . . . . . . . 12 |- ( x e. U_ b e. X C <-> E. b e. X x e. C )
3		disjiund.2	. . . . . . . . . . . . . . 15 |- ( b = d -> C = D )
4	3	eleq2d 2687	. . . . . . . . . . . . . 14 |- ( b = d -> ( x e. C <-> x e. D ) )
5	4	cbvrexv 3172	. . . . . . . . . . . . 13 |- ( E. b e. X x e. C <-> E. d e. X x e. D )
6		disjiund.4	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. A /\ x e. D ) -> a = c )
7	6	3exp 1264	. . . . . . . . . . . . . . . 16 |- ( ph -> ( x e. A -> ( x e. D -> a = c ) ) )
8	7	rexlimdvw 3034	. . . . . . . . . . . . . . 15 |- ( ph -> ( E. b e. W x e. A -> ( x e. D -> a = c ) ) )
9	8	imp 445	. . . . . . . . . . . . . 14 |- ( ( ph /\ E. b e. W x e. A ) -> ( x e. D -> a = c ) )
10	9	rexlimdvw 3034	. . . . . . . . . . . . 13 |- ( ( ph /\ E. b e. W x e. A ) -> ( E. d e. X x e. D -> a = c ) )
11	5, 10	syl5bi 232	. . . . . . . . . . . 12 |- ( ( ph /\ E. b e. W x e. A ) -> ( E. b e. X x e. C -> a = c ) )
12	2, 11	syl5bi 232	. . . . . . . . . . 11 |- ( ( ph /\ E. b e. W x e. A ) -> ( x e. U_ b e. X C -> a = c ) )
13	12	con3d 148	. . . . . . . . . 10 |- ( ( ph /\ E. b e. W x e. A ) -> ( -. a = c -> -. x e. U_ b e. X C ) )
14	13	impancom 456	. . . . . . . . 9 |- ( ( ph /\ -. a = c ) -> ( E. b e. W x e. A -> -. x e. U_ b e. X C ) )
15	1, 14	syl5bi 232	. . . . . . . 8 |- ( ( ph /\ -. a = c ) -> ( x e. U_ b e. W A -> -. x e. U_ b e. X C ) )
16	15	ralrimiv 2965	. . . . . . 7 |- ( ( ph /\ -. a = c ) -> A. x e. U_ b e. W A -. x e. U_ b e. X C )
17		disj 4017	. . . . . . 7 |- ( ( U_ b e. W A i^i U_ b e. X C ) = (/) <-> A. x e. U_ b e. W A -. x e. U_ b e. X C )
18	16, 17	sylibr 224	. . . . . 6 |- ( ( ph /\ -. a = c ) -> ( U_ b e. W A i^i U_ b e. X C ) = (/) )
19	18	ex 450	. . . . 5 |- ( ph -> ( -. a = c -> ( U_ b e. W A i^i U_ b e. X C ) = (/) ) )
20	19	orrd 393	. . . 4 |- ( ph -> ( a = c \/ ( U_ b e. W A i^i U_ b e. X C ) = (/) ) )
21	20	a1d 25	. . 3 |- ( ph -> ( ( a e. V /\ c e. V ) -> ( a = c \/ ( U_ b e. W A i^i U_ b e. X C ) = (/) ) ) )
22	21	ralrimivv 2970	. 2 |- ( ph -> A. a e. V A. c e. V ( a = c \/ ( U_ b e. W A i^i U_ b e. X C ) = (/) ) )
23		disjiund.3	. . 3 |- ( a = c -> W = X )
24		disjiund.1	. . 3 |- ( a = c -> A = C )
25	23, 24	disjiunb 4642	. 2 |- (Disj a e. V U_ b e. W A <-> A. a e. V A. c e. V ( a = c \/ ( U_ b e. W A i^i U_ b e. X C ) = (/) ) )
26	22, 25	sylibr 224	1 |- ( ph -> Disj a e. V U_ b e. W A )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   (/)c0 3915   U_ciun 4520  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-ral 2917  df-rex 2918  df-rmo 2920  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-iun 4522  df-disj 4621

This theorem is referenced by:  2wspiundisj  26856