Theorem disji2 4636

Description: Property of a disjoint collection: if B ( X ) = C and B ( Y ) = D, and X =/= Y, then C and D are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
disji.1	|- ( x = X -> B = C )
disji.2	|- ( x = Y -> B = D )

Assertion

Ref	Expression
disji2	|- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) /\ X =/= Y ) -> ( C i^i D ) = (/) )

Distinct variable groups:   x, A   x, C   x, D   x, X   x, Y

Allowed substitution hint:   B( x)

Proof of Theorem disji2

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

Step	Hyp	Ref	Expression
1		df-ne 2795	. . 3 |- ( X =/= Y <-> -. X = Y )
2		disjors 4635	. . . . . 6 |- (Disj x e. A B <-> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) )
3		eqeq1 2626	. . . . . . . 8 |- ( y = X -> ( y = z <-> X = z ) )
4		nfcv 2764	. . . . . . . . . . 11 |- F/_ x X
5		nfcv 2764	. . . . . . . . . . 11 |- F/_ x C
6		disji.1	. . . . . . . . . . 11 |- ( x = X -> B = C )
7	4, 5, 6	csbhypf 3552	. . . . . . . . . 10 |- ( y = X -> [_ y / x ]_ B = C )
8	7	ineq1d 3813	. . . . . . . . 9 |- ( y = X -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = ( C i^i [_ z / x ]_ B ) )
9	8	eqeq1d 2624	. . . . . . . 8 |- ( y = X -> ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) <-> ( C i^i [_ z / x ]_ B ) = (/) ) )
10	3, 9	orbi12d 746	. . . . . . 7 |- ( y = X -> ( ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) <-> ( X = z \/ ( C i^i [_ z / x ]_ B ) = (/) ) ) )
11		eqeq2 2633	. . . . . . . 8 |- ( z = Y -> ( X = z <-> X = Y ) )
12		nfcv 2764	. . . . . . . . . . 11 |- F/_ x Y
13		nfcv 2764	. . . . . . . . . . 11 |- F/_ x D
14		disji.2	. . . . . . . . . . 11 |- ( x = Y -> B = D )
15	12, 13, 14	csbhypf 3552	. . . . . . . . . 10 |- ( z = Y -> [_ z / x ]_ B = D )
16	15	ineq2d 3814	. . . . . . . . 9 |- ( z = Y -> ( C i^i [_ z / x ]_ B ) = ( C i^i D ) )
17	16	eqeq1d 2624	. . . . . . . 8 |- ( z = Y -> ( ( C i^i [_ z / x ]_ B ) = (/) <-> ( C i^i D ) = (/) ) )
18	11, 17	orbi12d 746	. . . . . . 7 |- ( z = Y -> ( ( X = z \/ ( C i^i [_ z / x ]_ B ) = (/) ) <-> ( X = Y \/ ( C i^i D ) = (/) ) ) )
19	10, 18	rspc2v 3322	. . . . . 6 |- ( ( X e. A /\ Y e. A ) -> ( A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) -> ( X = Y \/ ( C i^i D ) = (/) ) ) )
20	2, 19	syl5bi 232	. . . . 5 |- ( ( X e. A /\ Y e. A ) -> (Disj x e. A B -> ( X = Y \/ ( C i^i D ) = (/) ) ) )
21	20	impcom 446	. . . 4 |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) ) -> ( X = Y \/ ( C i^i D ) = (/) ) )
22	21	ord 392	. . 3 |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) ) -> ( -. X = Y -> ( C i^i D ) = (/) ) )
23	1, 22	syl5bi 232	. 2 |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) ) -> ( X =/= Y -> ( C i^i D ) = (/) ) )
24	23	3impia 1261	1 |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) /\ X =/= Y ) -> ( C i^i D ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   i^i cin 3573   (/)c0 3915  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-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by:  disji  4637  disjxiun  4649  disjxiunOLD  4650  voliunlem1  23318  disjf1  39369