Step |
Hyp |
Ref |
Expression |
0 |
|
clines |
|- Lines |
1 |
|
vk |
|- k |
2 |
|
cvv |
|- _V |
3 |
|
vs |
|- s |
4 |
|
vq |
|- q |
5 |
|
catm |
|- Atoms |
6 |
1
|
cv |
|- k |
7 |
6 5
|
cfv |
|- ( Atoms ` k ) |
8 |
|
vr |
|- r |
9 |
4
|
cv |
|- q |
10 |
8
|
cv |
|- r |
11 |
9 10
|
wne |
|- q =/= r |
12 |
3
|
cv |
|- s |
13 |
|
vp |
|- p |
14 |
13
|
cv |
|- p |
15 |
|
cple |
|- le |
16 |
6 15
|
cfv |
|- ( le ` k ) |
17 |
|
cjn |
|- join |
18 |
6 17
|
cfv |
|- ( join ` k ) |
19 |
9 10 18
|
co |
|- ( q ( join ` k ) r ) |
20 |
14 19 16
|
wbr |
|- p ( le ` k ) ( q ( join ` k ) r ) |
21 |
20 13 7
|
crab |
|- { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } |
22 |
12 21
|
wceq |
|- s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } |
23 |
11 22
|
wa |
|- ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) |
24 |
23 8 7
|
wrex |
|- E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) |
25 |
24 4 7
|
wrex |
|- E. q e. ( Atoms ` k ) E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) |
26 |
25 3
|
cab |
|- { s | E. q e. ( Atoms ` k ) E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) } |
27 |
1 2 26
|
cmpt |
|- ( k e. _V |-> { s | E. q e. ( Atoms ` k ) E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) } ) |
28 |
0 27
|
wceq |
|- Lines = ( k e. _V |-> { s | E. q e. ( Atoms ` k ) E. r e. ( Atoms ` k ) ( q =/= r /\ s = { p e. ( Atoms ` k ) | p ( le ` k ) ( q ( join ` k ) r ) } ) } ) |