Step |
Hyp |
Ref |
Expression |
1 |
|
iscgra.p |
|- P = ( Base ` G ) |
2 |
|
iscgra.i |
|- I = ( Itv ` G ) |
3 |
|
iscgra.k |
|- K = ( hlG ` G ) |
4 |
|
iscgra.g |
|- ( ph -> G e. TarskiG ) |
5 |
|
iscgra.a |
|- ( ph -> A e. P ) |
6 |
|
iscgra.b |
|- ( ph -> B e. P ) |
7 |
|
iscgra.c |
|- ( ph -> C e. P ) |
8 |
|
iscgra.d |
|- ( ph -> D e. P ) |
9 |
|
iscgra.e |
|- ( ph -> E e. P ) |
10 |
|
iscgra.f |
|- ( ph -> F e. P ) |
11 |
|
cgrahl1.2 |
|- ( ph -> <" A B C "> ( cgrA ` G ) <" D E F "> ) |
12 |
|
simpllr |
|- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> x e. P ) |
13 |
8
|
ad3antrrr |
|- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> D e. P ) |
14 |
9
|
ad3antrrr |
|- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> E e. P ) |
15 |
4
|
ad3antrrr |
|- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> G e. TarskiG ) |
16 |
|
simpr2 |
|- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> x ( K ` E ) D ) |
17 |
1 2 3 12 13 14 15 16
|
hlne2 |
|- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> D =/= E ) |
18 |
17
|
necomd |
|- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> E =/= D ) |
19 |
1 2 3 4 5 6 7 8 9 10
|
iscgra |
|- ( ph -> ( <" A B C "> ( cgrA ` G ) <" D E F "> <-> E. x e. P E. y e. P ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) ) |
20 |
11 19
|
mpbid |
|- ( ph -> E. x e. P E. y e. P ( <" A B C "> ( cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) |
21 |
18 20
|
r19.29vva |
|- ( ph -> E =/= D ) |