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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 ≠ ∅ → ∃ 𝑥 ∈ 𝐴 ( 𝑥 ∩ 𝐴 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		zfregfr	⊢ E Fr 𝐴
2		epfrs	⊢ ( ( E Fr 𝐴 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ( 𝑥 ∩ 𝐴 ) = ∅ )
3	1 2	mpan	⊢ ( 𝐴 ≠ ∅ → ∃ 𝑥 ∈ 𝐴 ( 𝑥 ∩ 𝐴 ) = ∅ )