Theorem disjuniel 33070

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 5017	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
2	1	ineq1i 4162	. 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 33069	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) = ∅)
8	2, 7	eqtrid 2807	1 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867  ∪ ciun 4951  Disj wdisj 5070

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-sn 4585  df-uni 4868  df-iun 4953  df-disj 5071

This theorem is used by:  carsgclctunlem1  34828