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		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
2	1	mo4	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) )
3		r2al	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) )
4	2 3	bitr4i	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 )