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Metamath Proof Explorer

Theorem hlimreui

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

		Ref	Expression
	Assertion	hlimreui	$⊢ \exists x \in H F \underset{v}{⇝} x \leftrightarrow \exists! x \in H F \underset{v}{⇝} x$

Proof

Step	Hyp	Ref	Expression
1		hlimuni	$⊢ (F \underset{v}{⇝} x \land F \underset{v}{⇝} y) \to x = y$
2	1	rgen2w	$⊢ \forall x \in H \forall y \in H ((F \underset{v}{⇝} x \land F \underset{v}{⇝} y) \to x = y)$
3	2	biantru	$⊢ \exists x \in H F \underset{v}{⇝} x \leftrightarrow (\exists x \in H F \underset{v}{⇝} x \land \forall x \in H \forall y \in H ((F \underset{v}{⇝} x \land F \underset{v}{⇝} y) \to x = y))$
4		breq2	$⊢ x = y \to (F \underset{v}{⇝} x \leftrightarrow F \underset{v}{⇝} y)$
5	4	reu4	$⊢ \exists! x \in H F \underset{v}{⇝} x \leftrightarrow (\exists x \in H F \underset{v}{⇝} x \land \forall x \in H \forall y \in H ((F \underset{v}{⇝} x \land F \underset{v}{⇝} y) \to x = y))$
6	3 5	bitr4i	$⊢ \exists x \in H F \underset{v}{⇝} x \leftrightarrow \exists! x \in H F \underset{v}{⇝} x$