Theorem ringbn0

Description: The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010) (Revised by AV, 25-Aug-2011)

	Ref	Expression
Hypothesis	ringbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	ringbn0	⊢ ( 𝐺 ∈ Ring → 𝐵 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		ringbn0.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ringgrp	⊢ ( 𝐺 ∈ Ring → 𝐺 ∈ Grp )
3	1	grpbn0	⊢ ( 𝐺 ∈ Grp → 𝐵 ≠ ∅ )
4	2 3	syl	⊢ ( 𝐺 ∈ Ring → 𝐵 ≠ ∅ )