Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003) (Proof shortened by Andrew Salmon, 26-Jun-2011) (Proof shortened by OpenAI, 25-Mar-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | prsspw.1 | |- A e. _V |
|
prsspw.2 | |- B e. _V |
||
Assertion | prsspw | |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prsspw.1 | |- A e. _V |
|
2 | prsspw.2 | |- B e. _V |
|
3 | prsspwg | |- ( ( A e. _V /\ B e. _V ) -> ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) ) |
|
4 | 1 2 3 | mp2an | |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) |