Theorem mndbn0

Description: The base set of a monoid is not empty. Statement in Lang p. 3. (Contributed by AV, 29-Dec-2023)

	Ref	Expression
Hypothesis	mndbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	mndbn0	⊢ ( 𝐺 ∈ Mnd → 𝐵 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		mndbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
3	1 2	mndidcl	⊢ ( 𝐺 ∈ Mnd → ( 0g ‘ 𝐺 ) ∈ 𝐵 )
4	3	ne0d	⊢ ( 𝐺 ∈ Mnd → 𝐵 ≠ ∅ )