Theorem disjuniel 32682

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 5002	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
2	1	ineq1i 4157	. 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 32681	. 2 ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶) = ∅)
8	2, 7	eqtrid 2784	1 ⊢ (𝜑 → (∪ 𝐵 ∩ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  ∪ cuni 4851  ∪ ciun 4934  Disj wdisj 5053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rmo 3343  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-sn 4569  df-uni 4852  df-iun 4936  df-disj 5054

This theorem is referenced by:  carsgclctunlem1  34477