Step |
Hyp |
Ref |
Expression |
1 |
|
choccl |
|- ( A e. CH -> ( _|_ ` A ) e. CH ) |
2 |
|
chsscon3 |
|- ( ( ( _|_ ` A ) e. CH /\ B e. CH ) -> ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ ( _|_ ` ( _|_ ` A ) ) ) ) |
3 |
1 2
|
sylan |
|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ ( _|_ ` ( _|_ ` A ) ) ) ) |
4 |
|
ococ |
|- ( A e. CH -> ( _|_ ` ( _|_ ` A ) ) = A ) |
5 |
4
|
adantr |
|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` A ) ) = A ) |
6 |
5
|
sseq2d |
|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` B ) C_ ( _|_ ` ( _|_ ` A ) ) <-> ( _|_ ` B ) C_ A ) ) |
7 |
3 6
|
bitrd |
|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` A ) C_ B <-> ( _|_ ` B ) C_ A ) ) |