Step |
Hyp |
Ref |
Expression |
1 |
|
ne0i |
|- ( 1o e. A -> A =/= (/) ) |
2 |
|
1on |
|- 1o e. On |
3 |
2
|
onirri |
|- -. 1o e. 1o |
4 |
|
eleq2 |
|- ( A = 1o -> ( 1o e. A <-> 1o e. 1o ) ) |
5 |
3 4
|
mtbiri |
|- ( A = 1o -> -. 1o e. A ) |
6 |
5
|
necon2ai |
|- ( 1o e. A -> A =/= 1o ) |
7 |
1 6
|
jca |
|- ( 1o e. A -> ( A =/= (/) /\ A =/= 1o ) ) |
8 |
|
el1o |
|- ( A e. 1o <-> A = (/) ) |
9 |
8
|
biimpi |
|- ( A e. 1o -> A = (/) ) |
10 |
9
|
necon3ai |
|- ( A =/= (/) -> -. A e. 1o ) |
11 |
|
nesym |
|- ( A =/= 1o <-> -. 1o = A ) |
12 |
11
|
biimpi |
|- ( A =/= 1o -> -. 1o = A ) |
13 |
10 12
|
anim12ci |
|- ( ( A =/= (/) /\ A =/= 1o ) -> ( -. 1o = A /\ -. A e. 1o ) ) |
14 |
|
pm4.56 |
|- ( ( -. 1o = A /\ -. A e. 1o ) <-> -. ( 1o = A \/ A e. 1o ) ) |
15 |
13 14
|
sylib |
|- ( ( A =/= (/) /\ A =/= 1o ) -> -. ( 1o = A \/ A e. 1o ) ) |
16 |
2
|
onordi |
|- Ord 1o |
17 |
|
ordtri2 |
|- ( ( Ord 1o /\ Ord A ) -> ( 1o e. A <-> -. ( 1o = A \/ A e. 1o ) ) ) |
18 |
16 17
|
mpan |
|- ( Ord A -> ( 1o e. A <-> -. ( 1o = A \/ A e. 1o ) ) ) |
19 |
15 18
|
imbitrrid |
|- ( Ord A -> ( ( A =/= (/) /\ A =/= 1o ) -> 1o e. A ) ) |
20 |
7 19
|
impbid2 |
|- ( Ord A -> ( 1o e. A <-> ( A =/= (/) /\ A =/= 1o ) ) ) |