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 ∈ Sℋ → ( 𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ ) )

Proof

Step	Hyp	Ref	Expression
1		sh0le	⊢ ( 𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴 )
2	1	biantrud	⊢ ( 𝐴 ∈ Sℋ → ( 𝐴 ⊆ 0ℋ ↔ ( 𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴 ) ) )
3		eqss	⊢ ( 𝐴 = 0ℋ ↔ ( 𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴 ) )
4	2 3	bitr4di	⊢ ( 𝐴 ∈ Sℋ → ( 𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ ) )