Step |
Hyp |
Ref |
Expression |
1 |
|
prunioo |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( ( A (,) B ) u. { A , B } ) = ( A [,] B ) ) |
2 |
|
uncom |
|- ( ( A (,) B ) u. { A , B } ) = ( { A , B } u. ( A (,) B ) ) |
3 |
1 2
|
eqtr3di |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( A [,] B ) = ( { A , B } u. ( A (,) B ) ) ) |
4 |
3
|
difeq1d |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( ( A [,] B ) \ ( A (,) B ) ) = ( ( { A , B } u. ( A (,) B ) ) \ ( A (,) B ) ) ) |
5 |
|
difun2 |
|- ( ( { A , B } u. ( A (,) B ) ) \ ( A (,) B ) ) = ( { A , B } \ ( A (,) B ) ) |
6 |
5
|
a1i |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( ( { A , B } u. ( A (,) B ) ) \ ( A (,) B ) ) = ( { A , B } \ ( A (,) B ) ) ) |
7 |
|
incom |
|- ( ( A (,) B ) i^i { A , B } ) = ( { A , B } i^i ( A (,) B ) ) |
8 |
|
iooinlbub |
|- ( ( A (,) B ) i^i { A , B } ) = (/) |
9 |
7 8
|
eqtr3i |
|- ( { A , B } i^i ( A (,) B ) ) = (/) |
10 |
|
disj3 |
|- ( ( { A , B } i^i ( A (,) B ) ) = (/) <-> { A , B } = ( { A , B } \ ( A (,) B ) ) ) |
11 |
9 10
|
mpbi |
|- { A , B } = ( { A , B } \ ( A (,) B ) ) |
12 |
11
|
eqcomi |
|- ( { A , B } \ ( A (,) B ) ) = { A , B } |
13 |
12
|
a1i |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( { A , B } \ ( A (,) B ) ) = { A , B } ) |
14 |
4 6 13
|
3eqtrd |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( ( A [,] B ) \ ( A (,) B ) ) = { A , B } ) |