Step |
Hyp |
Ref |
Expression |
1 |
|
1psubcl.a |
|- A = ( Atoms ` K ) |
2 |
|
1psubcl.c |
|- C = ( PSubCl ` K ) |
3 |
|
snssi |
|- ( Q e. A -> { Q } C_ A ) |
4 |
3
|
adantl |
|- ( ( K e. HL /\ Q e. A ) -> { Q } C_ A ) |
5 |
|
eqid |
|- ( _|_P ` K ) = ( _|_P ` K ) |
6 |
1 5
|
2polatN |
|- ( ( K e. HL /\ Q e. A ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` { Q } ) ) = { Q } ) |
7 |
1 5 2
|
ispsubclN |
|- ( K e. HL -> ( { Q } e. C <-> ( { Q } C_ A /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` { Q } ) ) = { Q } ) ) ) |
8 |
7
|
adantr |
|- ( ( K e. HL /\ Q e. A ) -> ( { Q } e. C <-> ( { Q } C_ A /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` { Q } ) ) = { Q } ) ) ) |
9 |
4 6 8
|
mpbir2and |
|- ( ( K e. HL /\ Q e. A ) -> { Q } e. C ) |