Step |
Hyp |
Ref |
Expression |
0 |
|
cer |
|- ~R |
1 |
|
vx |
|- x |
2 |
|
vy |
|- y |
3 |
1
|
cv |
|- x |
4 |
|
cnp |
|- P. |
5 |
4 4
|
cxp |
|- ( P. X. P. ) |
6 |
3 5
|
wcel |
|- x e. ( P. X. P. ) |
7 |
2
|
cv |
|- y |
8 |
7 5
|
wcel |
|- y e. ( P. X. P. ) |
9 |
6 8
|
wa |
|- ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) |
10 |
|
vz |
|- z |
11 |
|
vw |
|- w |
12 |
|
vv |
|- v |
13 |
|
vu |
|- u |
14 |
10
|
cv |
|- z |
15 |
11
|
cv |
|- w |
16 |
14 15
|
cop |
|- <. z , w >. |
17 |
3 16
|
wceq |
|- x = <. z , w >. |
18 |
12
|
cv |
|- v |
19 |
13
|
cv |
|- u |
20 |
18 19
|
cop |
|- <. v , u >. |
21 |
7 20
|
wceq |
|- y = <. v , u >. |
22 |
17 21
|
wa |
|- ( x = <. z , w >. /\ y = <. v , u >. ) |
23 |
|
cpp |
|- +P. |
24 |
14 19 23
|
co |
|- ( z +P. u ) |
25 |
15 18 23
|
co |
|- ( w +P. v ) |
26 |
24 25
|
wceq |
|- ( z +P. u ) = ( w +P. v ) |
27 |
22 26
|
wa |
|- ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) |
28 |
27 13
|
wex |
|- E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) |
29 |
28 12
|
wex |
|- E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) |
30 |
29 11
|
wex |
|- E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) |
31 |
30 10
|
wex |
|- E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) |
32 |
9 31
|
wa |
|- ( ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) ) |
33 |
32 1 2
|
copab |
|- { <. x , y >. | ( ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) ) } |
34 |
0 33
|
wceq |
|- ~R = { <. x , y >. | ( ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) ) } |