Step |
Hyp |
Ref |
Expression |
1 |
|
limom |
|- Lim _om |
2 |
|
ssel |
|- ( A C_ { x e. On | -. Lim x } -> ( _om e. A -> _om e. { x e. On | -. Lim x } ) ) |
3 |
|
limeq |
|- ( x = _om -> ( Lim x <-> Lim _om ) ) |
4 |
3
|
notbid |
|- ( x = _om -> ( -. Lim x <-> -. Lim _om ) ) |
5 |
4
|
elrab |
|- ( _om e. { x e. On | -. Lim x } <-> ( _om e. On /\ -. Lim _om ) ) |
6 |
5
|
simprbi |
|- ( _om e. { x e. On | -. Lim x } -> -. Lim _om ) |
7 |
2 6
|
syl6 |
|- ( A C_ { x e. On | -. Lim x } -> ( _om e. A -> -. Lim _om ) ) |
8 |
1 7
|
mt2i |
|- ( A C_ { x e. On | -. Lim x } -> -. _om e. A ) |
9 |
8
|
adantl |
|- ( ( Ord A /\ A C_ { x e. On | -. Lim x } ) -> -. _om e. A ) |
10 |
|
ordom |
|- Ord _om |
11 |
|
ordtri1 |
|- ( ( Ord A /\ Ord _om ) -> ( A C_ _om <-> -. _om e. A ) ) |
12 |
10 11
|
mpan2 |
|- ( Ord A -> ( A C_ _om <-> -. _om e. A ) ) |
13 |
12
|
adantr |
|- ( ( Ord A /\ A C_ { x e. On | -. Lim x } ) -> ( A C_ _om <-> -. _om e. A ) ) |
14 |
9 13
|
mpbird |
|- ( ( Ord A /\ A C_ { x e. On | -. Lim x } ) -> A C_ _om ) |