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Metamath Proof Explorer

Theorem hne0

Description: Hilbert space has a nonzero vector iff it is not trivial. (Contributed by NM, 24-Feb-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	hne0	$⊢ ℋ \neq 0_{ℋ} \leftrightarrow \exists x \in ℋ x \neq 0_{ℎ}$

Step	Hyp	Ref	Expression
1		helch	$⊢ ℋ \in C_{ℋ}$
2	1	chne0i	$⊢ ℋ \neq 0_{ℋ} \leftrightarrow \exists x \in ℋ x \neq 0_{ℎ}$