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Metamath Proof Explorer

Theorem in0

Description: The intersection of a class with the empty set is the empty set. Theorem 16 of Suppes p. 26. (Contributed by NM, 21-Jun-1993)

		Ref	Expression
	Assertion	in0	$⊢ A \cap \emptyset = \emptyset$

Step	Hyp	Ref	Expression
1		noel	$⊢ \neg x \in \emptyset$
2	1	bianfi	$⊢ x \in \emptyset \leftrightarrow (x \in A \land x \in \emptyset)$
3	2	bicomi	$⊢ (x \in A \land x \in \emptyset) \leftrightarrow x \in \emptyset$
4	3	ineqri	$⊢ A \cap \emptyset = \emptyset$