Theorem mosneq

Description: There exists at most one set whose singleton is equal to a given class. See also moeq . (Contributed by BJ, 24-Sep-2022)

	Ref	Expression
Assertion	mosneq	⊢ ∃* 𝑥 { 𝑥 } = 𝐴

Proof

Step	Hyp	Ref	Expression
1		eqtr3	⊢ ( ( { 𝑥 } = 𝐴 ∧ { 𝑦 } = 𝐴 ) → { 𝑥 } = { 𝑦 } )
2		vex	⊢ 𝑥 ∈ V
3	2	sneqr	⊢ ( { 𝑥 } = { 𝑦 } → 𝑥 = 𝑦 )
4	1 3	syl	⊢ ( ( { 𝑥 } = 𝐴 ∧ { 𝑦 } = 𝐴 ) → 𝑥 = 𝑦 )
5	4	gen2	⊢ ∀ 𝑥 ∀ 𝑦 ( ( { 𝑥 } = 𝐴 ∧ { 𝑦 } = 𝐴 ) → 𝑥 = 𝑦 )
6		sneq	⊢ ( 𝑥 = 𝑦 → { 𝑥 } = { 𝑦 } )
7	6	eqeq1d	⊢ ( 𝑥 = 𝑦 → ( { 𝑥 } = 𝐴 ↔ { 𝑦 } = 𝐴 ) )
8	7	mo4	⊢ ( ∃* 𝑥 { 𝑥 } = 𝐴 ↔ ∀ 𝑥 ∀ 𝑦 ( ( { 𝑥 } = 𝐴 ∧ { 𝑦 } = 𝐴 ) → 𝑥 = 𝑦 ) )
9	5 8	mpbir	⊢ ∃* 𝑥 { 𝑥 } = 𝐴