Theorem lbsex

Description: Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014)

	Ref	Expression
Hypothesis	lbsex.j	$$⊢J=\mathrm{LBasis}\left(W\right)$$
Assertion	lbsex	$$⊢W\in \mathrm{LVec}\to J\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		lbsex.j	$$⊢J=\mathrm{LBasis}\left(W\right)$$
2		axac3	$$⊢\mathrm{CHOICE}$$
3	1	lbsexg	$$⊢\left(\mathrm{CHOICE}\wedge W\in \mathrm{LVec}\right)\to J\ne \varnothing$$
4	2 3	mpan	$$⊢W\in \mathrm{LVec}\to J\ne \varnothing$$