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 = ( LBasis ‘ 𝑊 )
Assertion	lbsex	⊢ ( 𝑊 ∈ LVec → 𝐽 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		lbsex.j	⊢ 𝐽 = ( LBasis ‘ 𝑊 )
2		axac3	⊢ CHOICE
3	1	lbsexg	⊢ ( ( CHOICE ∧ 𝑊 ∈ LVec ) → 𝐽 ≠ ∅ )
4	2 3	mpan	⊢ ( 𝑊 ∈ LVec → 𝐽 ≠ ∅ )