Theorem disjif2 29394

Description: Property of a disjoint collection: if B ( x ) and B ( Y ) = D have a common element Z, then x = Y. (Contributed by Thierry Arnoux, 6-Apr-2017.)

Hypotheses

Ref	Expression
disjif2.1	|- F/_ x A
disjif2.2	|- F/_ x C
disjif2.3	|- ( x = Y -> B = C )

Assertion

Ref	Expression
disjif2	|- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) /\ ( Z e. B /\ Z e. C ) ) -> x = Y )

Distinct variable group:   x, Y

Allowed substitution hints:   A( x)   B( x)   C( x)   Z( x)

Proof of Theorem disjif2

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

Step	Hyp	Ref	Expression
1		inelcm 4032	. 2 |- ( ( Z e. B /\ Z e. C ) -> ( B i^i C ) =/= (/) )
2		disjif2.1	. . . . . . . 8 |- F/_ x A
3	2	disjorsf 29393	. . . . . . 7 |- (Disj x e. A B <-> A. y e. A A. z e. A ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) )
4		equequ1 1952	. . . . . . . . 9 |- ( y = x -> ( y = z <-> x = z ) )
5		csbeq1 3536	. . . . . . . . . . . 12 |- ( y = x -> [_ y / x ]_ B = [_ x / x ]_ B )
6		csbid 3541	. . . . . . . . . . . 12 |- [_ x / x ]_ B = B
7	5, 6	syl6eq 2672	. . . . . . . . . . 11 |- ( y = x -> [_ y / x ]_ B = B )
8	7	ineq1d 3813	. . . . . . . . . 10 |- ( y = x -> ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = ( B i^i [_ z / x ]_ B ) )
9	8	eqeq1d 2624	. . . . . . . . 9 |- ( y = x -> ( ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) <-> ( B i^i [_ z / x ]_ B ) = (/) ) )
10	4, 9	orbi12d 746	. . . . . . . 8 |- ( y = x -> ( ( y = z \/ ( [_ y / x ]_ B i^i [_ z / x ]_ B ) = (/) ) <-> ( x = z \/ ( B i^i [_ z / x ]_ B ) = (/) ) ) )
11		eqeq2 2633	. . . . . . . . 9 |- ( z = Y -> ( x = z <-> x = Y ) )
12		nfcv 2764	. . . . . . . . . . . 12 |- F/_ x Y
13		disjif2.2	. . . . . . . . . . . 12 |- F/_ x C
14		disjif2.3	. . . . . . . . . . . 12 |- ( x = Y -> B = C )
15	12, 13, 14	csbhypf 3552	. . . . . . . . . . 11 |- ( z = Y -> [_ z / x ]_ B = C )
16	15	ineq2d 3814	. . . . . . . . . 10 |- ( z = Y -> ( B i^i [_ z / x ]_ B ) = ( B i^i C ) )
17	16	eqeq1d 2624	. . . . . . . . 9 |- ( z = Y -> ( ( B i^i [_ z / x ]_ B ) = (/) <-> ( B i^i C ) = (/) ) )
18	11, 17	orbi12d 746	. . . . . . . 8 |- ( z = Y -> ( ( x = z \/ ( B i^i [_ z / x ]_ B ) = (/) ) <-> ( x = Y \/ ( B i^i C ) = (/) ) ) )
19	10, 18	rspc2v 3322	. . . . . . 7 |- ( ( 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 \/ ( B i^i C ) = (/) ) ) )
20	3, 19	syl5bi 232	. . . . . 6 |- ( ( x e. A /\ Y e. A ) -> (Disj x e. A B -> ( x = Y \/ ( B i^i C ) = (/) ) ) )
21	20	impcom 446	. . . . 5 |- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) ) -> ( x = Y \/ ( B i^i C ) = (/) ) )
22	21	ord 392	. . . 4 |- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) ) -> ( -. x = Y -> ( B i^i C ) = (/) ) )
23	22	necon1ad 2811	. . 3 |- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) ) -> ( ( B i^i C ) =/= (/) -> x = Y ) )
24	23	3impia 1261	. 2 |- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) /\ ( B i^i C ) =/= (/) ) -> x = Y )
25	1, 24	syl3an3 1361	1 |- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) /\ ( Z e. B /\ Z e. C ) ) -> x = Y )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   F/_wnfc 2751   =/= 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-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:  disjabrexf  29396