Metamath Proof Explorer

Theorem disjin2

Description: If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 21-Jun-2020)

		Ref	Expression
	Assertion	disjin2	$⊢ \underset{x \in B}{Disj} C \to \underset{x \in B}{Disj} (A \cap C)$

Proof

Step	Hyp	Ref	Expression
1		elinel2	$⊢ y \in (A \cap C) \to y \in C$
2	1	rmoimi	$⊢ \exists^{} x \in B y \in C \to \exists^{} x \in B y \in (A \cap C)$
3	2	alimi	$⊢ \forall y \exists^{} x \in B y \in C \to \forall y \exists^{} x \in B y \in (A \cap C)$
4		df-disj	$⊢ \underset{x \in B}{Disj} C \leftrightarrow \forall y \exists^{*} x \in B y \in C$
5		df-disj	$⊢ \underset{x \in B}{Disj} (A \cap C) \leftrightarrow \forall y \exists^{*} x \in B y \in (A \cap C)$
6	3 4 5	3imtr4i	$⊢ \underset{x \in B}{Disj} C \to \underset{x \in B}{Disj} (A \cap C)$