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	$$⊢\exists yy\in x\to \exists y\left(y\in x\wedge \forall z\left(z\in y\to \neg z\in x\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfe1	$$⊢Ⅎy\exists y\left(y\in x\wedge \forall z\left(z\in y\to \neg z\in x\right)\right)$$
2		axregnd	$$⊢y\in x\to \exists y\left(y\in x\wedge \forall z\left(z\in y\to \neg z\in x\right)\right)$$
3	1 2	exlimi	$$⊢\exists yy\in x\to \exists y\left(y\in x\wedge \forall z\left(z\in y\to \neg z\in x\right)\right)$$