Step |
Hyp |
Ref |
Expression |
1 |
|
lhp2at.l |
|- .<_ = ( le ` K ) |
2 |
|
lhp2at.a |
|- A = ( Atoms ` K ) |
3 |
|
lhp2at.h |
|- H = ( LHyp ` K ) |
4 |
|
simprr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ p .<_ W ) ) -> p .<_ W ) |
5 |
1 2 3
|
lhpexle1 |
|- ( ( K e. HL /\ W e. H ) -> E. q e. A ( q .<_ W /\ q =/= p ) ) |
6 |
5
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ p .<_ W ) ) -> E. q e. A ( q .<_ W /\ q =/= p ) ) |
7 |
4 6
|
jca |
|- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ p .<_ W ) ) -> ( p .<_ W /\ E. q e. A ( q .<_ W /\ q =/= p ) ) ) |
8 |
|
necom |
|- ( p =/= q <-> q =/= p ) |
9 |
8
|
3anbi3i |
|- ( ( p .<_ W /\ q .<_ W /\ p =/= q ) <-> ( p .<_ W /\ q .<_ W /\ q =/= p ) ) |
10 |
|
3anass |
|- ( ( p .<_ W /\ q .<_ W /\ q =/= p ) <-> ( p .<_ W /\ ( q .<_ W /\ q =/= p ) ) ) |
11 |
9 10
|
bitri |
|- ( ( p .<_ W /\ q .<_ W /\ p =/= q ) <-> ( p .<_ W /\ ( q .<_ W /\ q =/= p ) ) ) |
12 |
11
|
rexbii |
|- ( E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) <-> E. q e. A ( p .<_ W /\ ( q .<_ W /\ q =/= p ) ) ) |
13 |
|
r19.42v |
|- ( E. q e. A ( p .<_ W /\ ( q .<_ W /\ q =/= p ) ) <-> ( p .<_ W /\ E. q e. A ( q .<_ W /\ q =/= p ) ) ) |
14 |
12 13
|
bitr2i |
|- ( ( p .<_ W /\ E. q e. A ( q .<_ W /\ q =/= p ) ) <-> E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) ) |
15 |
7 14
|
sylib |
|- ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ p .<_ W ) ) -> E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) ) |
16 |
1 2 3
|
lhpexle |
|- ( ( K e. HL /\ W e. H ) -> E. p e. A p .<_ W ) |
17 |
15 16
|
reximddv |
|- ( ( K e. HL /\ W e. H ) -> E. p e. A E. q e. A ( p .<_ W /\ q .<_ W /\ p =/= q ) ) |