Theorem grpbn0

Description: The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007)

	Ref	Expression
Hypothesis	grpbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	grpbn0	⊢ ( 𝐺 ∈ Grp → 𝐵 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		grpbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
3	1 2	grpidcl	⊢ ( 𝐺 ∈ Grp → ( 0g ‘ 𝐺 ) ∈ 𝐵 )
4	3	ne0d	⊢ ( 𝐺 ∈ Grp → 𝐵 ≠ ∅ )