Theorem disjif 29391

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, 30-Dec-2016.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x, A   x, Y

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

Proof of Theorem disjif

Step	Hyp	Ref	Expression
1		inelcm 4032	. 2 |- ( ( Z e. B /\ Z e. C ) -> ( B i^i C ) =/= (/) )
2		disjif.1	. . . . . 6 |- F/_ x C
3		disjif.2	. . . . . 6 |- ( x = Y -> B = C )
4	2, 3	disji2f 29390	. . . . 5 |- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) /\ x =/= Y ) -> ( B i^i C ) = (/) )
5	4	3expia 1267	. . . 4 |- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) ) -> ( x =/= Y -> ( B i^i C ) = (/) ) )
6	5	necon1d 2816	. . 3 |- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) ) -> ( ( B i^i C ) =/= (/) -> x = Y ) )
7	6	3impia 1261	. 2 |- ( (Disj x e. A B /\ ( x e. A /\ Y e. A ) /\ ( B i^i C ) =/= (/) ) -> x = Y )
8	1, 7	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   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   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:  disjabrex  29395