Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. RR* ) |
2 |
|
iccid |
|- ( A e. RR* -> ( A [,] A ) = { A } ) |
3 |
1 2
|
syl |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( A [,] A ) = { A } ) |
4 |
3
|
uneq1d |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( { A } u. ( A (,) B ) ) ) |
5 |
|
simp2 |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. RR* ) |
6 |
1
|
xrleidd |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A <_ A ) |
7 |
|
simp3 |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B ) |
8 |
|
df-icc |
|- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } ) |
9 |
|
df-ioo |
|- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } ) |
10 |
|
xrltnle |
|- ( ( A e. RR* /\ w e. RR* ) -> ( A < w <-> -. w <_ A ) ) |
11 |
|
df-ico |
|- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } ) |
12 |
|
xrlelttr |
|- ( ( w e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( w <_ A /\ A < B ) -> w < B ) ) |
13 |
|
xrltle |
|- ( ( A e. RR* /\ w e. RR* ) -> ( A < w -> A <_ w ) ) |
14 |
13
|
3adant1 |
|- ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) -> ( A < w -> A <_ w ) ) |
15 |
14
|
adantld |
|- ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) -> ( ( A <_ A /\ A < w ) -> A <_ w ) ) |
16 |
8 9 10 11 12 15
|
ixxun |
|- ( ( ( A e. RR* /\ A e. RR* /\ B e. RR* ) /\ ( A <_ A /\ A < B ) ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( A [,) B ) ) |
17 |
1 1 5 6 7 16
|
syl32anc |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( A [,) B ) ) |
18 |
4 17
|
eqtr3d |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) ) |