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Metamath Proof Explorer

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	$⊢ \exists x F \underset{v}{⇝} x \leftrightarrow \exists! x F \underset{v}{⇝} x$

Proof

Step	Hyp	Ref	Expression
1		hlimreui	$⊢ \exists x \in V F \underset{v}{⇝} x \leftrightarrow \exists! x \in V F \underset{v}{⇝} x$
2		rexv	$⊢ \exists x \in V F \underset{v}{⇝} x \leftrightarrow \exists x F \underset{v}{⇝} x$
3		reuv	$⊢ \exists! x \in V F \underset{v}{⇝} x \leftrightarrow \exists! x F \underset{v}{⇝} x$
4	1 2 3	3bitr3i	$⊢ \exists x F \underset{v}{⇝} x \leftrightarrow \exists! x F \underset{v}{⇝} x$