Step |
Hyp |
Ref |
Expression |
1 |
|
uncom |
|- ( ( A (,) B ) u. ( A [,] A ) ) = ( ( A [,] A ) u. ( A (,) B ) ) |
2 |
|
iccid |
|- ( A e. RR* -> ( A [,] A ) = { A } ) |
3 |
2
|
3ad2ant1 |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( A [,] A ) = { A } ) |
4 |
3
|
uneq2d |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. ( A [,] A ) ) = ( ( A (,) B ) u. { A } ) ) |
5 |
|
simp1 |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. RR* ) |
6 |
|
simp2 |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> B e. RR* ) |
7 |
|
xrleid |
|- ( A e. RR* -> A <_ A ) |
8 |
7
|
3ad2ant1 |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A <_ A ) |
9 |
|
simp3 |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A < B ) |
10 |
|
df-icc |
|- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } ) |
11 |
|
df-ioo |
|- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } ) |
12 |
|
xrltnle |
|- ( ( A e. RR* /\ w e. RR* ) -> ( A < w <-> -. w <_ A ) ) |
13 |
|
df-ico |
|- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } ) |
14 |
|
xrlelttr |
|- ( ( w e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( w <_ A /\ A < B ) -> w < B ) ) |
15 |
|
simpl1 |
|- ( ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) /\ ( A <_ A /\ A < w ) ) -> A e. RR* ) |
16 |
|
simpl3 |
|- ( ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) /\ ( A <_ A /\ A < w ) ) -> w e. RR* ) |
17 |
|
simprr |
|- ( ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) /\ ( A <_ A /\ A < w ) ) -> A < w ) |
18 |
15 16 17
|
xrltled |
|- ( ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) /\ ( A <_ A /\ A < w ) ) -> A <_ w ) |
19 |
18
|
ex |
|- ( ( A e. RR* /\ A e. RR* /\ w e. RR* ) -> ( ( A <_ A /\ A < w ) -> A <_ w ) ) |
20 |
10 11 12 13 14 19
|
ixxun |
|- ( ( ( A e. RR* /\ A e. RR* /\ B e. RR* ) /\ ( A <_ A /\ A < B ) ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( A [,) B ) ) |
21 |
5 5 6 8 9 20
|
syl32anc |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A [,] A ) u. ( A (,) B ) ) = ( A [,) B ) ) |
22 |
1 4 21
|
3eqtr3a |
|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { A } ) = ( A [,) B ) ) |