Theorem disji2 4075

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 4074	. . 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 2413	. . . . 5 |- ( y = X -> ( y =/= z <-> X =/= z ) )
3		nfcv 2372	. . . . . . . 8 |- F/_ x X
4		nfcv 2372	. . . . . . . 8 |- F/_ x C
5		disji.1	. . . . . . . 8 |- ( x = X -> B = C )
6	3, 4, 5	csbhypf 3163	. . . . . . 7 |- ( y = X -> [_ y / x ]_ B = C )
7	6	ineq1d 3404	. . . . . 6 |- ( y = X -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = ( C i^i [_ z / x ]_ B ) )
8	7	eqeq1d 2238	. . . . 5 |- ( y = X -> ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) <-> ( C i^i [_ z / x ]_ B ) = (/) ) )
9	2, 8	imbi12d 234	. . . 4 |- ( y = X -> ( ( y =/= z -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) <-> ( X =/= z -> ( C i^i [_ z / x ]_ B ) = (/) ) ) )
10		neeq2 2414	. . . . 5 |- ( z = Y -> ( X =/= z <-> X =/= Y ) )
11		nfcv 2372	. . . . . . . 8 |- F/_ x Y
12		nfcv 2372	. . . . . . . 8 |- F/_ x D
13		disji.2	. . . . . . . 8 |- ( x = Y -> B = D )
14	11, 12, 13	csbhypf 3163	. . . . . . 7 |- ( z = Y -> [_ z / x ]_ B = D )
15	14	ineq2d 3405	. . . . . 6 |- ( z = Y -> ( C i^i [_ z / x ]_ B ) = ( C i^i D ) )
16	15	eqeq1d 2238	. . . . 5 |- ( z = Y -> ( ( C i^i [_ z / x ]_ B ) = (/) <-> ( C i^i D ) = (/) ) )
17	10, 16	imbi12d 234	. . . 4 |- ( z = Y -> ( ( X =/= z -> ( C i^i [_ z / x ]_ B ) = (/) ) <-> ( X =/= Y -> ( C i^i D ) = (/) ) ) )
18	9, 17	rspc2v 2920	. . 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 281	. 2 |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) ) -> ( X =/= Y -> ( C i^i D ) = (/) ) )
20	19	3impia 1224	1 |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) /\ X =/= Y ) -> ( C i^i D ) = (/) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   [_csb 3124   i^i cin 3196   (/)c0 3491  Disj wdisj 4059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-in 3203  df-nul 3492  df-disj 4060

This theorem is referenced by: (None)