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Metamath Proof Explorer

Theorem nelaneq

Description: A class is not an element of and equal to a class at the same time. Variant of elneq analogously to elnotel and en2lp . (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022) (Proof shortened by TM, 31-Dec-2025) (Proof shortened by SN, 22-Apr-2026)

		Ref	Expression
	Assertion	nelaneq	\|- -. ( A e. B /\ A = B )

Proof

Step	Hyp	Ref	Expression
1		elirr	\|- -. A e. A
2		eleq2	\|- ( A = B -> ( A e. A <-> A e. B ) )
3	2	biimparc	\|- ( ( A e. B /\ A = B ) -> A e. A )
4	1 3	mto	\|- -. ( A e. B /\ A = B )