Step |
Hyp |
Ref |
Expression |
1 |
|
atl0le.b |
|- B = ( Base ` K ) |
2 |
|
atl0le.l |
|- .<_ = ( le ` K ) |
3 |
|
atl0le.z |
|- .0. = ( 0. ` K ) |
4 |
1 2 3
|
atl0le |
|- ( ( K e. AtLat /\ X e. B ) -> .0. .<_ X ) |
5 |
4
|
biantrud |
|- ( ( K e. AtLat /\ X e. B ) -> ( X .<_ .0. <-> ( X .<_ .0. /\ .0. .<_ X ) ) ) |
6 |
|
atlpos |
|- ( K e. AtLat -> K e. Poset ) |
7 |
6
|
adantr |
|- ( ( K e. AtLat /\ X e. B ) -> K e. Poset ) |
8 |
|
simpr |
|- ( ( K e. AtLat /\ X e. B ) -> X e. B ) |
9 |
1 3
|
atl0cl |
|- ( K e. AtLat -> .0. e. B ) |
10 |
9
|
adantr |
|- ( ( K e. AtLat /\ X e. B ) -> .0. e. B ) |
11 |
1 2
|
posasymb |
|- ( ( K e. Poset /\ X e. B /\ .0. e. B ) -> ( ( X .<_ .0. /\ .0. .<_ X ) <-> X = .0. ) ) |
12 |
7 8 10 11
|
syl3anc |
|- ( ( K e. AtLat /\ X e. B ) -> ( ( X .<_ .0. /\ .0. .<_ X ) <-> X = .0. ) ) |
13 |
5 12
|
bitrd |
|- ( ( K e. AtLat /\ X e. B ) -> ( X .<_ .0. <-> X = .0. ) ) |