Theorem chle0i

Description: No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chle0i	$$⊢A\subseteq 0_{\mathscr{H}}\leftrightarrow A=0_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chle0	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to \left(A\subseteq 0_{\mathscr{H}}\leftrightarrow A=0_{\mathscr{H}}\right)$$
3	1 2	ax-mp	$$⊢A\subseteq 0_{\mathscr{H}}\leftrightarrow A=0_{\mathscr{H}}$$