Theorem moel

Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018)

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

Proof

Step	Hyp	Ref	Expression
1		ralcom4	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) )
2		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) )
3	2	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) )
4		alcom	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) )
5		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
6	5	mo4	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) )
7		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ) )
8		impexp	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ) )
9	8	albii	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ) )
10	7 9	bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) )
11	10	albii	⊢ ( ∀ 𝑦 ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 = 𝑦 ) )
12	4 6 11	3bitr4i	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∀ 𝑦 ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐴 → 𝑥 = 𝑦 ) )
13	1 3 12	3bitr4ri	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑥 = 𝑦 )