Step |
Hyp |
Ref |
Expression |
1 |
|
nlim0 |
|- -. Lim (/) |
2 |
|
limeq |
|- ( A = (/) -> ( Lim A <-> Lim (/) ) ) |
3 |
1 2
|
mtbiri |
|- ( A = (/) -> -. Lim A ) |
4 |
3
|
necon2ai |
|- ( Lim A -> A =/= (/) ) |
5 |
|
nlim1 |
|- -. Lim 1o |
6 |
|
limeq |
|- ( A = 1o -> ( Lim A <-> Lim 1o ) ) |
7 |
5 6
|
mtbiri |
|- ( A = 1o -> -. Lim A ) |
8 |
7
|
necon2ai |
|- ( Lim A -> A =/= 1o ) |
9 |
|
limord |
|- ( Lim A -> Ord A ) |
10 |
|
ord1eln01 |
|- ( Ord A -> ( 1o e. A <-> ( A =/= (/) /\ A =/= 1o ) ) ) |
11 |
9 10
|
syl |
|- ( Lim A -> ( 1o e. A <-> ( A =/= (/) /\ A =/= 1o ) ) ) |
12 |
4 8 11
|
mpbir2and |
|- ( Lim A -> 1o e. A ) |