Theorem disjuniel 32757

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 5013	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
2	1	ineq1i 4166	. 2 ⊢ (∪ 𝐵 ∩ 𝐶) = (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶)
3		disjuniel.1	. . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥)
4		id 22	. . 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 32756	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) = ∅)
8	2, 7	eqtrid 2808	1 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  ∪ cuni 4862  ∪ ciun 4946  Disj wdisj 5064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rmo 3366  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-sn 4580  df-uni 4863  df-iun 4948  df-disj 5065

This theorem is referenced by:  carsgclctunlem1  34575