Theorem disjiun2 39226

Description: In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
disjiun2.1	|- ( ph -> Disj x e. A B )
disjiun2.2	|- ( ph -> C C_ A )
disjiun2.3	|- ( ph -> D e. ( A \ C ) )
disjiun2.4	|- ( x = D -> B = E )

Assertion

Ref	Expression
disjiun2	|- ( ph -> ( U_ x e. C B i^i E ) = (/) )

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

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

Proof of Theorem disjiun2

Step	Hyp	Ref	Expression
1		disjiun2.3	. . . 4 |- ( ph -> D e. ( A \ C ) )
2		disjiun2.4	. . . . 5 |- ( x = D -> B = E )
3	2	iunxsng 4602	. . . 4 |- ( D e. ( A \ C ) -> U_ x e. { D } B = E )
4	1, 3	syl 17	. . 3 |- ( ph -> U_ x e. { D } B = E )
5	4	ineq2d 3814	. 2 |- ( ph -> ( U_ x e. C B i^i U_ x e. { D } B ) = ( U_ x e. C B i^i E ) )
6		disjiun2.1	. . 3 |- ( ph -> Disj x e. A B )
7		disjiun2.2	. . 3 |- ( ph -> C C_ A )
8		eldifi 3732	. . . 4 |- ( D e. ( A \ C ) -> D e. A )
9		snssi 4339	. . . 4 |- ( D e. A -> { D } C_ A )
10	1, 8, 9	3syl 18	. . 3 |- ( ph -> { D } C_ A )
11	1	eldifbd 3587	. . . 4 |- ( ph -> -. D e. C )
12		disjsn 4246	. . . 4 |- ( ( C i^i { D } ) = (/) <-> -. D e. C )
13	11, 12	sylibr 224	. . 3 |- ( ph -> ( C i^i { D } ) = (/) )
14		disjiun 4640	. . 3 |- ( (Disj x e. A B /\ ( C C_ A /\ { D } C_ A /\ ( C i^i { D } ) = (/) ) ) -> ( U_ x e. C B i^i U_ x e. { D } B ) = (/) )
15	6, 7, 10, 13, 14	syl13anc 1328	. 2 |- ( ph -> ( U_ x e. C B i^i U_ x e. { D } B ) = (/) )
16	5, 15	eqtr3d 2658	1 |- ( ph -> ( U_ x e. C B i^i E ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   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-ne 2795  df-ral 2917  df-rex 2918  df-rmo 2920  df-v 3202  df-sbc 3436  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-iun 4522  df-disj 4621

This theorem is referenced by:  caratheodorylem1  40740