Theorem disji2 3917

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		disjnims 3916	. . 3 |- (Disj x e. A B -> A. y e. A A. z e. A ( y =/= z -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) )
2		neeq1 2319	. . . . 5 |- ( y = X -> ( y =/= z <-> X =/= z ) )
3		nfcv 2279	. . . . . . . 8 |- F/_ x X
4		nfcv 2279	. . . . . . . 8 |- F/_ x C
5		disji.1	. . . . . . . 8 |- ( x = X -> B = C )
6	3, 4, 5	csbhypf 3033	. . . . . . 7 |- ( y = X -> [_ y / x ]_ B = C )
7	6	ineq1d 3271	. . . . . 6 |- ( y = X -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = ( C i^i [_ z / x ]_ B ) )
8	7	eqeq1d 2146	. . . . 5 |- ( y = X -> ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) <-> ( C i^i [_ z / x ]_ B ) = (/) ) )
9	2, 8	imbi12d 233	. . . 4 |- ( y = X -> ( ( y =/= z -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) <-> ( X =/= z -> ( C i^i [_ z / x ]_ B ) = (/) ) ) )
10		neeq2 2320	. . . . 5 |- ( z = Y -> ( X =/= z <-> X =/= Y ) )
11		nfcv 2279	. . . . . . . 8 |- F/_ x Y
12		nfcv 2279	. . . . . . . 8 |- F/_ x D
13		disji.2	. . . . . . . 8 |- ( x = Y -> B = D )
14	11, 12, 13	csbhypf 3033	. . . . . . 7 |- ( z = Y -> [_ z / x ]_ B = D )
15	14	ineq2d 3272	. . . . . 6 |- ( z = Y -> ( C i^i [_ z / x ]_ B ) = ( C i^i D ) )
16	15	eqeq1d 2146	. . . . 5 |- ( z = Y -> ( ( C i^i [_ z / x ]_ B ) = (/) <-> ( C i^i D ) = (/) ) )
17	10, 16	imbi12d 233	. . . 4 |- ( z = Y -> ( ( X =/= z -> ( C i^i [_ z / x ]_ B ) = (/) ) <-> ( X =/= Y -> ( C i^i D ) = (/) ) ) )
18	9, 17	rspc2v 2797	. . 3 |- ( ( 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 ) = (/) ) ) )
19	1, 18	mpan9 279	. 2 |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) ) -> ( X =/= Y -> ( C i^i D ) = (/) ) )
20	19	3impia 1178	1 |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) /\ X =/= Y ) -> ( C i^i D ) = (/) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2306   A.wral 2414   [_csb 2998   i^i cin 3065   (/)c0 3358  Disj wdisj 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-in 3072  df-nul 3359  df-disj 3902

This theorem is referenced by: (None)