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Metamath Proof Explorer

Theorem disjr

Description: Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011)

		Ref	Expression
	Assertion	disjr	$⊢ A \cap B = \emptyset \leftrightarrow \forall x \in B \neg x \in A$

Step	Hyp	Ref	Expression
1		ineqcom	$⊢ A \cap B = \emptyset \leftrightarrow B \cap A = \emptyset$
2		disj	$⊢ B \cap A = \emptyset \leftrightarrow \forall x \in B \neg x \in A$
3	1 2	bitri	$⊢ A \cap B = \emptyset \leftrightarrow \forall x \in B \neg x \in A$