Description: The empty set is not a signed real. (Contributed by NM, 25-Aug-1995) (Revised by Mario Carneiro, 10-Jul-2014) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | 0nsr | |- -. (/) e. R. |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid | |- (/) = (/) |
|
2 | enrer | |- ~R Er ( P. X. P. ) |
|
3 | erdm | |- ( ~R Er ( P. X. P. ) -> dom ~R = ( P. X. P. ) ) |
|
4 | 2 3 | ax-mp | |- dom ~R = ( P. X. P. ) |
5 | elqsn0 | |- ( ( dom ~R = ( P. X. P. ) /\ (/) e. ( ( P. X. P. ) /. ~R ) ) -> (/) =/= (/) ) |
|
6 | 4 5 | mpan | |- ( (/) e. ( ( P. X. P. ) /. ~R ) -> (/) =/= (/) ) |
7 | df-nr | |- R. = ( ( P. X. P. ) /. ~R ) |
|
8 | 6 7 | eleq2s | |- ( (/) e. R. -> (/) =/= (/) ) |
9 | 8 | necon2bi | |- ( (/) = (/) -> -. (/) e. R. ) |
10 | 1 9 | ax-mp | |- -. (/) e. R. |