Metamath Proof Explorer


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 ≠ ∅ → ∃ 𝑥𝐴 ( 𝑥𝐴 ) = ∅ )

Proof

Step Hyp Ref Expression
1 zfregfr E Fr 𝐴
2 epfrs ( ( E Fr 𝐴𝐴 ≠ ∅ ) → ∃ 𝑥𝐴 ( 𝑥𝐴 ) = ∅ )
3 1 2 mpan ( 𝐴 ≠ ∅ → ∃ 𝑥𝐴 ( 𝑥𝐴 ) = ∅ )