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Metamath Proof Explorer

Theorem nelaneqOLD

Description: Obsolete version of nelaneq as of 22-Apr-2026. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022) (Proof shortened by TM, 31-Dec-2025) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	nelaneqOLD	$⊢ \neg (A \in B \land A = B)$

Proof

Step	Hyp	Ref	Expression
1		elirr	$⊢ \neg A \in A$
2		eleq2	$⊢ A = B \to (A \in A \leftrightarrow A \in B)$
3	1 2	mtbii	$⊢ A = B \to \neg A \in B$
4	3	con2i	$⊢ A \in B \to \neg A = B$
5		imnan	$⊢ (A \in B \to \neg A = B) \leftrightarrow \neg (A \in B \land A = B)$
6	4 5	mpbi	$⊢ \neg (A \in B \land A = B)$