Theorem prmidlssidl

Description: Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024)

	Ref	Expression
Assertion	prmidlssidl	⊢ ( 𝑅 ∈ Ring → ( PrmIdeal ‘ 𝑅 ) ⊆ ( LIdeal ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		prmidlidl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑖 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑖 ∈ ( LIdeal ‘ 𝑅 ) )
2	1	ex	⊢ ( 𝑅 ∈ Ring → ( 𝑖 ∈ ( PrmIdeal ‘ 𝑅 ) → 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ) )
3	2	ssrdv	⊢ ( 𝑅 ∈ Ring → ( PrmIdeal ‘ 𝑅 ) ⊆ ( LIdeal ‘ 𝑅 ) )