Theorem subrngss

Description: A subring is a subset. (Contributed by AV, 14-Feb-2025)

	Ref	Expression
Hypothesis	subrngss.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion	subrngss	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝐴 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		subrngss.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
2	1	issubrng	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ↔ ( 𝑅 ∈ Rng ∧ ( 𝑅 ↾s 𝐴 ) ∈ Rng ∧ 𝐴 ⊆ 𝐵 ) )
3	2	simp3bi	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝐴 ⊆ 𝐵 )