This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem shle0

Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004) (New usage is discouraged.)

		Ref	Expression
	Assertion	shle0	$⊢ A \in S_{ℋ} \to (A \subseteq 0_{ℋ} \leftrightarrow A = 0_{ℋ})$

Step	Hyp	Ref	Expression
1		sh0le	$⊢ A \in S_{ℋ} \to 0_{ℋ} \subseteq A$
2	1	biantrud	$⊢ A \in S_{ℋ} \to (A \subseteq 0_{ℋ} \leftrightarrow (A \subseteq 0_{ℋ} \land 0_{ℋ} \subseteq A))$
3		eqss	$⊢ A = 0_{ℋ} \leftrightarrow (A \subseteq 0_{ℋ} \land 0_{ℋ} \subseteq A)$
4	2 3	bitr4di	$⊢ A \in S_{ℋ} \to (A \subseteq 0_{ℋ} \leftrightarrow A = 0_{ℋ})$