Description: A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998) (Revised by BJ, 12-Feb-2022)
Ref | Expression | ||
---|---|---|---|
Assertion | relsnopg | |- ( ( A e. V /\ B e. W ) -> Rel { <. A , B >. } ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelvvg | |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. ( _V X. _V ) ) |
|
2 | opex | |- <. A , B >. e. _V |
|
3 | relsng | |- ( <. A , B >. e. _V -> ( Rel { <. A , B >. } <-> <. A , B >. e. ( _V X. _V ) ) ) |
|
4 | 2 3 | mp1i | |- ( ( A e. V /\ B e. W ) -> ( Rel { <. A , B >. } <-> <. A , B >. e. ( _V X. _V ) ) ) |
5 | 1 4 | mpbird | |- ( ( A e. V /\ B e. W ) -> Rel { <. A , B >. } ) |