Step |
Hyp |
Ref |
Expression |
1 |
|
eldif |
|- ( A e. ( On \ 2o ) <-> ( A e. On /\ -. A e. 2o ) ) |
2 |
|
1on |
|- 1o e. On |
3 |
|
ontri1 |
|- ( ( A e. On /\ 1o e. On ) -> ( A C_ 1o <-> -. 1o e. A ) ) |
4 |
|
onsssuc |
|- ( ( A e. On /\ 1o e. On ) -> ( A C_ 1o <-> A e. suc 1o ) ) |
5 |
|
df-2o |
|- 2o = suc 1o |
6 |
5
|
eleq2i |
|- ( A e. 2o <-> A e. suc 1o ) |
7 |
4 6
|
bitr4di |
|- ( ( A e. On /\ 1o e. On ) -> ( A C_ 1o <-> A e. 2o ) ) |
8 |
3 7
|
bitr3d |
|- ( ( A e. On /\ 1o e. On ) -> ( -. 1o e. A <-> A e. 2o ) ) |
9 |
2 8
|
mpan2 |
|- ( A e. On -> ( -. 1o e. A <-> A e. 2o ) ) |
10 |
9
|
con1bid |
|- ( A e. On -> ( -. A e. 2o <-> 1o e. A ) ) |
11 |
10
|
pm5.32i |
|- ( ( A e. On /\ -. A e. 2o ) <-> ( A e. On /\ 1o e. A ) ) |
12 |
1 11
|
bitri |
|- ( A e. ( On \ 2o ) <-> ( A e. On /\ 1o e. A ) ) |