Theorem disjuniel 29410

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
disjuniel.1	|- ( ph -> Disj x e. A x )
disjuniel.2	|- ( ph -> B C_ A )
disjuniel.3	|- ( ph -> C e. ( A \ B ) )

Assertion

Ref	Expression
disjuniel	|- ( ph -> ( U. B i^i C ) = (/) )

Distinct variable groups:   x, A   x, B   x, C

Allowed substitution hint:   ph( x)

Proof of Theorem disjuniel

Step	Hyp	Ref	Expression
1		uniiun 4573	. . 3 |- U. B = U_ x e. B x
2	1	ineq1i 3810	. 2 |- ( U. B i^i C ) = ( U_ x e. B x i^i C )
3		disjuniel.1	. . 3 |- ( ph -> Disj x e. A x )
4		id 22	. . 3 |- ( x = C -> x = C )
5		disjuniel.2	. . 3 |- ( ph -> B C_ A )
6		disjuniel.3	. . 3 |- ( ph -> C e. ( A \ B ) )
7	3, 4, 5, 6	disjiunel 29409	. 2 |- ( ph -> ( U_ x e. B x i^i C ) = (/) )
8	2, 7	syl5eq 2668	1 |- ( ph -> ( U. B i^i C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   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-uni 4437  df-iun 4522  df-disj 4621

This theorem is referenced by:  carsgclctunlem1  30379