Theorem disjiunel 29409

Description: A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020.)

Hypotheses

Ref	Expression
disjiunel.1	|- ( ph -> Disj x e. A B )
disjiunel.2	|- ( x = Y -> B = D )
disjiunel.3	|- ( ph -> E C_ A )
disjiunel.4	|- ( ph -> Y e. ( A \ E ) )

Assertion

Ref	Expression
disjiunel	|- ( ph -> ( U_ x e. E B i^i D ) = (/) )

Distinct variable groups:   x, A   x, D   x, E   x, Y

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem disjiunel

Step	Hyp	Ref	Expression
1		disjiunel.3	. . . . 5 |- ( ph -> E C_ A )
2		disjiunel.4	. . . . . . 7 |- ( ph -> Y e. ( A \ E ) )
3	2	eldifad 3586	. . . . . 6 |- ( ph -> Y e. A )
4	3	snssd 4340	. . . . 5 |- ( ph -> { Y } C_ A )
5	1, 4	unssd 3789	. . . 4 |- ( ph -> ( E u. { Y } ) C_ A )
6		disjiunel.1	. . . 4 |- ( ph -> Disj x e. A B )
7		disjss1 4626	. . . 4 |- ( ( E u. { Y } ) C_ A -> (Disj x e. A B -> Disj x e. ( E u. { Y } ) B ) )
8	5, 6, 7	sylc 65	. . 3 |- ( ph -> Disj x e. ( E u. { Y } ) B )
9	2	eldifbd 3587	. . . 4 |- ( ph -> -. Y e. E )
10		disjiunel.2	. . . . 5 |- ( x = Y -> B = D )
11	10	disjunsn 29407	. . . 4 |- ( ( Y e. A /\ -. Y e. E ) -> (Disj x e. ( E u. { Y } ) B <-> (Disj x e. E B /\ ( U_ x e. E B i^i D ) = (/) ) ) )
12	3, 9, 11	syl2anc 693	. . 3 |- ( ph -> (Disj x e. ( E u. { Y } ) B <-> (Disj x e. E B /\ ( U_ x e. E B i^i D ) = (/) ) ) )
13	8, 12	mpbid 222	. 2 |- ( ph -> (Disj x e. E B /\ ( U_ x e. E B i^i D ) = (/) ) )
14	13	simprd 479	1 |- ( ph -> ( U_ x e. E B i^i D ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520  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-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-iun 4522  df-disj 4621

This theorem is referenced by:  disjuniel  29410