Description: The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of TakeutiZaring p. 96. This is Metamath 100 proof #52. (Contributed by NM, 29-Jan-2004) (Proof shortened by Mario Carneiro, 1-Jul-2015)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | pw2en.1 | |- A e. _V |
|
| Assertion | pw2en | |- ~P A ~~ ( 2o ^m A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pw2en.1 | |- A e. _V |
|
| 2 | pw2eng | |- ( A e. _V -> ~P A ~~ ( 2o ^m A ) ) |
|
| 3 | 1 2 | ax-mp | |- ~P A ~~ ( 2o ^m A ) |