Theorem moeq

Description: There exists at most one set equal to a given class. (Contributed by NM, 8-Mar-1995) Shorten combined proofs of moeq and eueq . (Proof shortened by BJ, 24-Sep-2022)

	Ref	Expression
Assertion	moeq	⊢ ∃* 𝑥 𝑥 = 𝐴

Proof

Step	Hyp	Ref	Expression
1		eqtr3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ) → 𝑥 = 𝑦 )
2	1	gen2	⊢ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ) → 𝑥 = 𝑦 )
3		eqeq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐴 ↔ 𝑦 = 𝐴 ) )
4	3	mo4	⊢ ( ∃* 𝑥 𝑥 = 𝐴 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐴 ) → 𝑥 = 𝑦 ) )
5	2 4	mpbir	⊢ ∃* 𝑥 𝑥 = 𝐴