| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ftc2re.e |
|- E = ( C (,) D ) |
| 2 |
|
ftc2re.a |
|- ( ph -> A e. E ) |
| 3 |
|
ftc2re.b |
|- ( ph -> B e. E ) |
| 4 |
2 1
|
eleqtrdi |
|- ( ph -> A e. ( C (,) D ) ) |
| 5 |
|
eliooxr |
|- ( A e. ( C (,) D ) -> ( C e. RR* /\ D e. RR* ) ) |
| 6 |
4 5
|
syl |
|- ( ph -> ( C e. RR* /\ D e. RR* ) ) |
| 7 |
6
|
simpld |
|- ( ph -> C e. RR* ) |
| 8 |
6
|
simprd |
|- ( ph -> D e. RR* ) |
| 9 |
|
eliooord |
|- ( A e. ( C (,) D ) -> ( C < A /\ A < D ) ) |
| 10 |
4 9
|
syl |
|- ( ph -> ( C < A /\ A < D ) ) |
| 11 |
10
|
simpld |
|- ( ph -> C < A ) |
| 12 |
3 1
|
eleqtrdi |
|- ( ph -> B e. ( C (,) D ) ) |
| 13 |
|
eliooord |
|- ( B e. ( C (,) D ) -> ( C < B /\ B < D ) ) |
| 14 |
12 13
|
syl |
|- ( ph -> ( C < B /\ B < D ) ) |
| 15 |
14
|
simprd |
|- ( ph -> B < D ) |
| 16 |
|
iccssioo |
|- ( ( ( C e. RR* /\ D e. RR* ) /\ ( C < A /\ B < D ) ) -> ( A [,] B ) C_ ( C (,) D ) ) |
| 17 |
7 8 11 15 16
|
syl22anc |
|- ( ph -> ( A [,] B ) C_ ( C (,) D ) ) |
| 18 |
17 1
|
sseqtrrdi |
|- ( ph -> ( A [,] B ) C_ E ) |