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	biantru	⊢ ( 𝑦 = 𝐴 ↔ ( 𝑦 = 𝐴 ∧ 𝑦 ∈ { 𝑧 ∣ ⊤ } ) )
4	3	exbii	⊢ ( ∃ 𝑦 𝑦 = 𝐴 ↔ ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝑦 ∈ { 𝑧 ∣ ⊤ } ) )
5		dfclel	⊢ ( 𝐴 ∈ { 𝑧 ∣ ⊤ } ↔ ∃ 𝑦 ( 𝑦 = 𝐴 ∧ 𝑦 ∈ { 𝑧 ∣ ⊤ } ) )
6	4 5	bitr4i	⊢ ( ∃ 𝑦 𝑦 = 𝐴 ↔ 𝐴 ∈ { 𝑧 ∣ ⊤ } )
7		vextru	⊢ 𝑥 ∈ { 𝑧 ∣ ⊤ }
8	7	biantru	⊢ ( 𝑥 = 𝐴 ↔ ( 𝑥 = 𝐴 ∧ 𝑥 ∈ { 𝑧 ∣ ⊤ } ) )
9	8	exbii	⊢ ( ∃ 𝑥 𝑥 = 𝐴 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ { 𝑧 ∣ ⊤ } ) )
10		dfclel	⊢ ( 𝐴 ∈ { 𝑧 ∣ ⊤ } ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ { 𝑧 ∣ ⊤ } ) )
11	9 10	bitr4i	⊢ ( ∃ 𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ { 𝑧 ∣ ⊤ } )
12	6 11	bitr4i	⊢ ( ∃ 𝑦 𝑦 = 𝐴 ↔ ∃ 𝑥 𝑥 = 𝐴 )
13	1 12	sylib	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )