Theorem zfregs

Description: The strong form of the Axiom of Regularity, which does not require that A be a set. Axiom 6' of TakeutiZaring p. 21. See also epfrs . (Contributed by NM, 17-Sep-2003)

	Ref	Expression
Assertion	zfregs	$$⊢A\ne \varnothing \to \exists x\in Ax\cap A=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		zfregfr	$$⊢\mathrm{E}\mathrm{Fr}A$$
2		epfrs	$$⊢\left(\mathrm{E}\mathrm{Fr}A\wedge A\ne \varnothing \right)\to \exists x\in Ax\cap A=\varnothing$$
3	1 2	mpan	$$⊢A\ne \varnothing \to \exists x\in Ax\cap A=\varnothing$$