Metamath Proof Explorer

Theorem prmidlssidl

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

Ref Expression
Assertion prmidlssidl 
|- ( R e. Ring -> ( PrmIdeal ` R ) C_ ( LIdeal ` R ) )

Proof

Step Hyp Ref Expression
1 prmidlidl 
 |-  ( ( R e. Ring /\ i e. ( PrmIdeal ` R ) ) -> i e. ( LIdeal ` R ) )
2 1 ex 
 |-  ( R e. Ring -> ( i e. ( PrmIdeal ` R ) -> i e. ( LIdeal ` R ) ) )
3 2 ssrdv 
 |-  ( R e. Ring -> ( PrmIdeal ` R ) C_ ( LIdeal ` R ) )