Theorem hlimeui

Description: The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999) (Proof shortened by Mario Carneiro, 14-May-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	hlimeui	⊢ ( ∃ 𝑥 𝐹 ⇝𝑣 𝑥 ↔ ∃! 𝑥 𝐹 ⇝𝑣 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		hlimreui	⊢ ( ∃ 𝑥 ∈ V 𝐹 ⇝𝑣 𝑥 ↔ ∃! 𝑥 ∈ V 𝐹 ⇝𝑣 𝑥 )
2		rexv	⊢ ( ∃ 𝑥 ∈ V 𝐹 ⇝𝑣 𝑥 ↔ ∃ 𝑥 𝐹 ⇝𝑣 𝑥 )
3		reuv	⊢ ( ∃! 𝑥 ∈ V 𝐹 ⇝𝑣 𝑥 ↔ ∃! 𝑥 𝐹 ⇝𝑣 𝑥 )
4	1 2 3	3bitr3i	⊢ ( ∃ 𝑥 𝐹 ⇝𝑣 𝑥 ↔ ∃! 𝑥 𝐹 ⇝𝑣 𝑥 )