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Metamath Proof Explorer

Theorem disjuniel

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)

Ref Expression
Hypotheses disjuniel.1 ⊢ ( 𝜑 → Disj 𝑥 ∈ 𝐴 𝑥 )
disjuniel.2 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
disjuniel.3 ⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 ∖ 𝐵 ) )
Assertion disjuniel ( 𝜑 → ( ∪ 𝐵 ∩ 𝐶 ) = ∅ )

Proof

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