Theorem lssle0

Description: No subspace is smaller than the zero subspace. ( shle0 analog.) (Contributed by NM, 20-Apr-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lss0cl.z	$$⊢\underset{\dot{}}{0}=0_W$$
	lss0cl.s	$$⊢S=\mathrm{LSubSp}\left(W\right)$$
Assertion	lssle0	$$⊢\left(W\in \mathrm{LMod}\wedge X\in S\right)\to \left(X\subseteq \left\{\underset{\dot{}}{0}\right\}\leftrightarrow X=\left\{\underset{\dot{}}{0}\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		lss0cl.z	$$⊢\underset{\dot{}}{0}=0_W$$
2		lss0cl.s	$$⊢S=\mathrm{LSubSp}\left(W\right)$$
3	1 2	lss0ss	$$⊢\left(W\in \mathrm{LMod}\wedge X\in S\right)\to \left\{\underset{\dot{}}{0}\right\}\subseteq X$$
4	3	biantrud	$$⊢\left(W\in \mathrm{LMod}\wedge X\in S\right)\to \left(X\subseteq \left\{\underset{\dot{}}{0}\right\}\leftrightarrow \left(X\subseteq \left\{\underset{\dot{}}{0}\right\}\wedge \left\{\underset{\dot{}}{0}\right\}\subseteq X\right)\right)$$
5		eqss	$$⊢X=\left\{\underset{\dot{}}{0}\right\}\leftrightarrow \left(X\subseteq \left\{\underset{\dot{}}{0}\right\}\wedge \left\{\underset{\dot{}}{0}\right\}\subseteq X\right)$$
6	4 5	bitr4di	$$⊢\left(W\in \mathrm{LMod}\wedge X\in S\right)\to \left(X\subseteq \left\{\underset{\dot{}}{0}\right\}\leftrightarrow X=\left\{\underset{\dot{}}{0}\right\}\right)$$