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Metamath Proof Explorer

Theorem zfcndreg

Description: Axiom of Regularity ax-reg , reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-Aug-2003) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	zfcndreg	$⊢ \exists y y \in x \to \exists y (y \in x \land \forall z (z \in y \to \neg z \in x))$

Proof

Step	Hyp	Ref	Expression
1		nfe1	$⊢ Ⅎ y \exists y (y \in x \land \forall z (z \in y \to \neg z \in x))$
2		axregnd	$⊢ y \in x \to \exists y (y \in x \land \forall z (z \in y \to \neg z \in x))$
3	1 2	exlimi	$⊢ \exists y y \in x \to \exists y (y \in x \land \forall z (z \in y \to \neg z \in x))$