Step |
Hyp |
Ref |
Expression |
1 |
|
umgr2adedgwlk.e |
|- E = ( Edg ` G ) |
2 |
|
umgr2adedgwlk.i |
|- I = ( iEdg ` G ) |
3 |
|
umgr2adedgwlk.f |
|- F = <" J K "> |
4 |
|
umgr2adedgwlk.p |
|- P = <" A B C "> |
5 |
|
umgr2adedgwlk.g |
|- ( ph -> G e. UMGraph ) |
6 |
|
umgr2adedgwlk.a |
|- ( ph -> ( { A , B } e. E /\ { B , C } e. E ) ) |
7 |
|
umgr2adedgwlk.j |
|- ( ph -> ( I ` J ) = { A , B } ) |
8 |
|
umgr2adedgwlk.k |
|- ( ph -> ( I ` K ) = { B , C } ) |
9 |
|
umgr2adedgspth.n |
|- ( ph -> A =/= C ) |
10 |
|
3anass |
|- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) <-> ( G e. UMGraph /\ ( { A , B } e. E /\ { B , C } e. E ) ) ) |
11 |
5 6 10
|
sylanbrc |
|- ( ph -> ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) ) |
12 |
1
|
umgr2adedgwlklem |
|- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( ( A =/= B /\ B =/= C ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) ) |
13 |
11 12
|
syl |
|- ( ph -> ( ( A =/= B /\ B =/= C ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) ) |
14 |
13
|
simprd |
|- ( ph -> ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) |
15 |
13
|
simpld |
|- ( ph -> ( A =/= B /\ B =/= C ) ) |
16 |
|
ssid |
|- { A , B } C_ { A , B } |
17 |
16 7
|
sseqtrrid |
|- ( ph -> { A , B } C_ ( I ` J ) ) |
18 |
|
ssid |
|- { B , C } C_ { B , C } |
19 |
18 8
|
sseqtrrid |
|- ( ph -> { B , C } C_ ( I ` K ) ) |
20 |
17 19
|
jca |
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) ) |
21 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
22 |
|
fveq2 |
|- ( K = J -> ( I ` K ) = ( I ` J ) ) |
23 |
22
|
eqcoms |
|- ( J = K -> ( I ` K ) = ( I ` J ) ) |
24 |
23
|
eqeq1d |
|- ( J = K -> ( ( I ` K ) = { B , C } <-> ( I ` J ) = { B , C } ) ) |
25 |
|
eqtr2 |
|- ( ( ( I ` J ) = { B , C } /\ ( I ` J ) = { A , B } ) -> { B , C } = { A , B } ) |
26 |
25
|
ex |
|- ( ( I ` J ) = { B , C } -> ( ( I ` J ) = { A , B } -> { B , C } = { A , B } ) ) |
27 |
24 26
|
syl6bi |
|- ( J = K -> ( ( I ` K ) = { B , C } -> ( ( I ` J ) = { A , B } -> { B , C } = { A , B } ) ) ) |
28 |
27
|
com13 |
|- ( ( I ` J ) = { A , B } -> ( ( I ` K ) = { B , C } -> ( J = K -> { B , C } = { A , B } ) ) ) |
29 |
7 8 28
|
sylc |
|- ( ph -> ( J = K -> { B , C } = { A , B } ) ) |
30 |
|
eqcom |
|- ( { B , C } = { A , B } <-> { A , B } = { B , C } ) |
31 |
|
prcom |
|- { B , C } = { C , B } |
32 |
31
|
eqeq2i |
|- ( { A , B } = { B , C } <-> { A , B } = { C , B } ) |
33 |
30 32
|
bitri |
|- ( { B , C } = { A , B } <-> { A , B } = { C , B } ) |
34 |
21 1
|
umgrpredgv |
|- ( ( G e. UMGraph /\ { A , B } e. E ) -> ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) ) |
35 |
34
|
simpld |
|- ( ( G e. UMGraph /\ { A , B } e. E ) -> A e. ( Vtx ` G ) ) |
36 |
35
|
ex |
|- ( G e. UMGraph -> ( { A , B } e. E -> A e. ( Vtx ` G ) ) ) |
37 |
21 1
|
umgrpredgv |
|- ( ( G e. UMGraph /\ { B , C } e. E ) -> ( B e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) |
38 |
37
|
simprd |
|- ( ( G e. UMGraph /\ { B , C } e. E ) -> C e. ( Vtx ` G ) ) |
39 |
38
|
ex |
|- ( G e. UMGraph -> ( { B , C } e. E -> C e. ( Vtx ` G ) ) ) |
40 |
36 39
|
anim12d |
|- ( G e. UMGraph -> ( ( { A , B } e. E /\ { B , C } e. E ) -> ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) ) |
41 |
5 6 40
|
sylc |
|- ( ph -> ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) ) |
42 |
|
preqr1g |
|- ( ( A e. ( Vtx ` G ) /\ C e. ( Vtx ` G ) ) -> ( { A , B } = { C , B } -> A = C ) ) |
43 |
41 42
|
syl |
|- ( ph -> ( { A , B } = { C , B } -> A = C ) ) |
44 |
|
eqneqall |
|- ( A = C -> ( A =/= C -> J =/= K ) ) |
45 |
43 9 44
|
syl6ci |
|- ( ph -> ( { A , B } = { C , B } -> J =/= K ) ) |
46 |
33 45
|
syl5bi |
|- ( ph -> ( { B , C } = { A , B } -> J =/= K ) ) |
47 |
29 46
|
syld |
|- ( ph -> ( J = K -> J =/= K ) ) |
48 |
|
neqne |
|- ( -. J = K -> J =/= K ) |
49 |
47 48
|
pm2.61d1 |
|- ( ph -> J =/= K ) |
50 |
4 3 14 15 20 21 2 49 9
|
2spthd |
|- ( ph -> F ( SPaths ` G ) P ) |