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Metamath Proof Explorer

Theorem issetft

Description: Closed theorem form of isset that does not require x and A to be distinct. Extracted from the proof of vtoclgft . (Contributed by Wolf Lammen, 9-Apr-2025)

		Ref	Expression
	Assertion	issetft	⊢ ( Ⅎ 𝑥 𝐴 → ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		isset	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑦 𝑦 = 𝐴 )
2		cbvexeqsetf	⊢ ( Ⅎ 𝑥 𝐴 → ( ∃ 𝑥 𝑥 = 𝐴 ↔ ∃ 𝑦 𝑦 = 𝐴 ) )
3	1 2	bitr4id	⊢ ( Ⅎ 𝑥 𝐴 → ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 ) )