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Metamath Proof Explorer

Theorem issetf

Description: A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011) (Revised by Mario Carneiro, 10-Oct-2016)

		Ref	Expression
	Hypothesis	issetf.1	⊢ Ⅎ 𝑥 𝐴
	Assertion	issetf	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		issetf.1	⊢ Ⅎ 𝑥 𝐴
2		issetft	⊢ ( Ⅎ 𝑥 𝐴 → ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )