Theorem elisset

Description: An element of a class exists. Use elissetv instead when sufficient (for instance in usages where x is a dummy variable). (Contributed by NM, 1-May-1995) Reduce dependencies on axioms. (Revised by BJ, 29-Apr-2019)

	Ref	Expression
Assertion	elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		elissetv	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑦 𝑦 = 𝐴 )
2		vextru	⊢ 𝑦 ∈ { 𝑧 ∣ ⊤ }
3	2	issetlem	⊢ ( 𝐴 ∈ { 𝑧 ∣ ⊤ } ↔ ∃ 𝑦 𝑦 = 𝐴 )
4		vextru	⊢ 𝑥 ∈ { 𝑧 ∣ ⊤ }
5	4	issetlem	⊢ ( 𝐴 ∈ { 𝑧 ∣ ⊤ } ↔ ∃ 𝑥 𝑥 = 𝐴 )
6	3 5	bitr3i	⊢ ( ∃ 𝑦 𝑦 = 𝐴 ↔ ∃ 𝑥 𝑥 = 𝐴 )
7	1 6	sylib	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )