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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	⊢ ( ∃ 𝑦 𝑦 ∈ 𝑥 → ∃ 𝑦 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nfe1	⊢ Ⅎ 𝑦 ∃ 𝑦 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) )
2		axregnd	⊢ ( 𝑦 ∈ 𝑥 → ∃ 𝑦 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) ) )
3	1 2	exlimi	⊢ ( ∃ 𝑦 𝑦 ∈ 𝑥 → ∃ 𝑦 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) ) )