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

Proof

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