Theorem moel

Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018) Avoid ax-11 . (Revised by Wolf Lammen, 23-Nov-2024)

	Ref	Expression
Assertion	moel	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		19.21v	⊢ ( ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ) )
2		impexp	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ) )
3	2	albii	⊢ ( ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ) )
4		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) )
5	4	imbi2i	⊢ ( ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ) )
6	1 3 5	3bitr4i	⊢ ( ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 ) )
7	6	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 ) )
8		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
9	8	mo4	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) )
10		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 ) )
11	7 9 10	3bitr4i	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 )