Step |
Hyp |
Ref |
Expression |
1 |
|
swoer.1 |
|- R = ( ( X X. X ) \ ( .< u. `' .< ) ) |
2 |
|
opelxpi |
|- ( ( A e. X /\ B e. X ) -> <. A , B >. e. ( X X. X ) ) |
3 |
|
df-br |
|- ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) ) |
4 |
2 3
|
sylibr |
|- ( ( A e. X /\ B e. X ) -> A ( X X. X ) B ) |
5 |
1
|
breqi |
|- ( A R B <-> A ( ( X X. X ) \ ( .< u. `' .< ) ) B ) |
6 |
|
brdif |
|- ( A ( ( X X. X ) \ ( .< u. `' .< ) ) B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) ) |
7 |
5 6
|
bitri |
|- ( A R B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) ) |
8 |
7
|
baib |
|- ( A ( X X. X ) B -> ( A R B <-> -. A ( .< u. `' .< ) B ) ) |
9 |
4 8
|
syl |
|- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. A ( .< u. `' .< ) B ) ) |
10 |
|
brun |
|- ( A ( .< u. `' .< ) B <-> ( A .< B \/ A `' .< B ) ) |
11 |
|
brcnvg |
|- ( ( A e. X /\ B e. X ) -> ( A `' .< B <-> B .< A ) ) |
12 |
11
|
orbi2d |
|- ( ( A e. X /\ B e. X ) -> ( ( A .< B \/ A `' .< B ) <-> ( A .< B \/ B .< A ) ) ) |
13 |
10 12
|
bitrid |
|- ( ( A e. X /\ B e. X ) -> ( A ( .< u. `' .< ) B <-> ( A .< B \/ B .< A ) ) ) |
14 |
13
|
notbid |
|- ( ( A e. X /\ B e. X ) -> ( -. A ( .< u. `' .< ) B <-> -. ( A .< B \/ B .< A ) ) ) |
15 |
9 14
|
bitrd |
|- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) ) |