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Metamath Proof Explorer

Theorem disj1

Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993)

		Ref	Expression
	Assertion	disj1	$⊢ A \cap B = \emptyset \leftrightarrow \forall x (x \in A \to \neg x \in B)$

Step	Hyp	Ref	Expression
1		disj	$⊢ A \cap B = \emptyset \leftrightarrow \forall x \in A \neg x \in B$
2		df-ral	$⊢ \forall x \in A \neg x \in B \leftrightarrow \forall x (x \in A \to \neg x \in B)$
3	1 2	bitri	$⊢ A \cap B = \emptyset \leftrightarrow \forall x (x \in A \to \neg x \in B)$