Step |
Hyp |
Ref |
Expression |
1 |
|
prunioo |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( ( A (,) B ) u. { A , B } ) = ( A [,] B ) ) |
2 |
1
|
eleq2d |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( C e. ( ( A (,) B ) u. { A , B } ) <-> C e. ( A [,] B ) ) ) |
3 |
2
|
biimpar |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C e. ( A [,] B ) ) -> C e. ( ( A (,) B ) u. { A , B } ) ) |
4 |
|
elun |
|- ( C e. ( ( A (,) B ) u. { A , B } ) <-> ( C e. ( A (,) B ) \/ C e. { A , B } ) ) |
5 |
|
elprg |
|- ( C e. ( A [,] B ) -> ( C e. { A , B } <-> ( C = A \/ C = B ) ) ) |
6 |
5
|
orbi2d |
|- ( C e. ( A [,] B ) -> ( ( C e. ( A (,) B ) \/ C e. { A , B } ) <-> ( C e. ( A (,) B ) \/ ( C = A \/ C = B ) ) ) ) |
7 |
4 6
|
syl5bb |
|- ( C e. ( A [,] B ) -> ( C e. ( ( A (,) B ) u. { A , B } ) <-> ( C e. ( A (,) B ) \/ ( C = A \/ C = B ) ) ) ) |
8 |
7
|
adantl |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C e. ( A [,] B ) ) -> ( C e. ( ( A (,) B ) u. { A , B } ) <-> ( C e. ( A (,) B ) \/ ( C = A \/ C = B ) ) ) ) |
9 |
3 8
|
mpbid |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C e. ( A [,] B ) ) -> ( C e. ( A (,) B ) \/ ( C = A \/ C = B ) ) ) |
10 |
|
3orass |
|- ( ( C e. ( A (,) B ) \/ C = A \/ C = B ) <-> ( C e. ( A (,) B ) \/ ( C = A \/ C = B ) ) ) |
11 |
|
3orcoma |
|- ( ( C e. ( A (,) B ) \/ C = A \/ C = B ) <-> ( C = A \/ C e. ( A (,) B ) \/ C = B ) ) |
12 |
10 11
|
bitr3i |
|- ( ( C e. ( A (,) B ) \/ ( C = A \/ C = B ) ) <-> ( C = A \/ C e. ( A (,) B ) \/ C = B ) ) |
13 |
9 12
|
sylib |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C e. ( A [,] B ) ) -> ( C = A \/ C e. ( A (,) B ) \/ C = B ) ) |
14 |
|
lbicc2 |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) ) |
15 |
14
|
adantr |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C = A ) -> A e. ( A [,] B ) ) |
16 |
|
eleq1 |
|- ( C = A -> ( C e. ( A [,] B ) <-> A e. ( A [,] B ) ) ) |
17 |
16
|
adantl |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C = A ) -> ( C e. ( A [,] B ) <-> A e. ( A [,] B ) ) ) |
18 |
15 17
|
mpbird |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C = A ) -> C e. ( A [,] B ) ) |
19 |
|
ioossicc |
|- ( A (,) B ) C_ ( A [,] B ) |
20 |
19
|
sseli |
|- ( C e. ( A (,) B ) -> C e. ( A [,] B ) ) |
21 |
20
|
adantl |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C e. ( A (,) B ) ) -> C e. ( A [,] B ) ) |
22 |
|
ubicc2 |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) ) |
23 |
22
|
adantr |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C = B ) -> B e. ( A [,] B ) ) |
24 |
|
eleq1 |
|- ( C = B -> ( C e. ( A [,] B ) <-> B e. ( A [,] B ) ) ) |
25 |
24
|
adantl |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C = B ) -> ( C e. ( A [,] B ) <-> B e. ( A [,] B ) ) ) |
26 |
23 25
|
mpbird |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ C = B ) -> C e. ( A [,] B ) ) |
27 |
18 21 26
|
3jaodan |
|- ( ( ( A e. RR* /\ B e. RR* /\ A <_ B ) /\ ( C = A \/ C e. ( A (,) B ) \/ C = B ) ) -> C e. ( A [,] B ) ) |
28 |
13 27
|
impbida |
|- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> ( C e. ( A [,] B ) <-> ( C = A \/ C e. ( A (,) B ) \/ C = B ) ) ) |