Theorem disjuniel 32923

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	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥)
disjuniel.2	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
disjuniel.3	⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))

Assertion

Ref	Expression
disjuniel	⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅)

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem disjuniel

Step	Hyp	Ref	Expression
1		uniiun 5024	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
2	1	ineq1i 4170	. 2 ⊢ (∪ 𝐵 ∩ 𝐶) = (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶)
3		disjuniel.1	. . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥)
4		id 23	. . 3 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶)
5		disjuniel.2	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
6		disjuniel.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))
7	3, 4, 5, 6	disjiunel 32922	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) = ∅)
8	2, 7	eqtrid 2810	1 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  ∪ cuni 4873  ∪ ciun 4957  Disj wdisj 5077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-sn 4591  df-uni 4874  df-iun 4959  df-disj 5078

This theorem is referenced by:  carsgclctunlem1  34688