Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of Mendelson p. 255. (Contributed by NM, 17-Dec-2003) (Proof shortened by AV, 17-Jul-2022)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | map1 | |- ( A e. V -> ( 1o ^m A ) ~~ 1o ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df1o2 | |- 1o = { (/) } |
|
| 2 | 1 | oveq1i | |- ( 1o ^m A ) = ( { (/) } ^m A ) |
| 3 | 0ex | |- (/) e. _V |
|
| 4 | snmapen1 | |- ( ( (/) e. _V /\ A e. V ) -> ( { (/) } ^m A ) ~~ 1o ) |
|
| 5 | 3 4 | mpan | |- ( A e. V -> ( { (/) } ^m A ) ~~ 1o ) |
| 6 | 2 5 | eqbrtrid | |- ( A e. V -> ( 1o ^m A ) ~~ 1o ) |