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Metamath Proof Explorer

Theorem nelne1

Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012) (Proof shortened by Wolf Lammen, 14-May-2023)

		Ref	Expression
	Assertion	nelne1	$⊢ (A \in B \land \neg A \in C) \to B \neq C$

Proof

Step	Hyp	Ref	Expression
1		nelneq2	$⊢ (A \in B \land \neg A \in C) \to \neg B = C$
2	1	neqned	$⊢ (A \in B \land \neg A \in C) \to B \neq C$