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Metamath Proof Explorer

Theorem nd3

Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002)

		Ref	Expression
	Assertion	nd3	$⊢ \forall x x = y \to \neg \forall z x \in y$

Step	Hyp	Ref	Expression
1		elirrv	$⊢ \neg x \in x$
2		elequ2	$⊢ x = y \to (x \in x \leftrightarrow x \in y)$
3	1 2	mtbii	$⊢ x = y \to \neg x \in y$
4	3	sps	$⊢ \forall x x = y \to \neg x \in y$
5		sp	$⊢ \forall z x \in y \to x \in y$
6	4 5	nsyl	$⊢ \forall x x = y \to \neg \forall z x \in y$