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Theorem usgra2pth0 32145
Description: In a undirected simply graph, there is a path of length 2 if and only if there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
Assertion
Ref Expression
usgra2pth0  |-  ( V USGrph  E  ->  ( ( F ( V Paths  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
( 0..^ 2 )
-1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
Distinct variable groups:    x, E, y, z    x, F, y, z    x, P, y, z    x, V, y, z

Proof of Theorem usgra2pth0
StepHypRef Expression
1 usgra2pth 32144 . 2  |-  ( V USGrph  E  ->  ( ( F ( V Paths  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
( 0..^ 2 )
-1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. z  e.  ( V  \  { x } ) E. y  e.  ( V  \  { x ,  z } ) ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
2 r19.42v 3021 . . . . . . . . 9  |-  ( E. y  e.  ( V 
\  { x ,  z } ) ( z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) )  <->  ( z  =/=  x  /\  E. y  e.  ( V  \  {
x ,  z } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
3 rexdifpr 32090 . . . . . . . . 9  |-  ( E. y  e.  ( V 
\  { x ,  z } ) ( z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) )  <->  E. y  e.  V  ( y  =/=  x  /\  y  =/=  z  /\  ( z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
42, 3bitr3i 251 . . . . . . . 8  |-  ( ( z  =/=  x  /\  E. y  e.  ( V 
\  { x ,  z } ) ( ( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) )  <->  E. y  e.  V  ( y  =/=  x  /\  y  =/=  z  /\  ( z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
54rexbii 2969 . . . . . . 7  |-  ( E. z  e.  V  ( z  =/=  x  /\  E. y  e.  ( V 
\  { x ,  z } ) ( ( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) )  <->  E. z  e.  V  E. y  e.  V  ( y  =/=  x  /\  y  =/=  z  /\  ( z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
6 rexcom 3028 . . . . . . 7  |-  ( E. z  e.  V  E. y  e.  V  (
y  =/=  x  /\  y  =/=  z  /\  (
z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )  <->  E. y  e.  V  E. z  e.  V  ( y  =/=  x  /\  y  =/=  z  /\  ( z  =/=  x  /\  (
( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) ) ) )
7 df-3an 975 . . . . . . . . . . 11  |-  ( ( y  =/=  x  /\  y  =/=  z  /\  (
z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )  <->  ( (
y  =/=  x  /\  y  =/=  z )  /\  ( z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
8 anass 649 . . . . . . . . . . 11  |-  ( ( ( ( y  =/=  x  /\  y  =/=  z )  /\  z  =/=  x )  /\  (
( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) )  <->  ( ( y  =/=  x  /\  y  =/=  z )  /\  (
z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
9 anass 649 . . . . . . . . . . . 12  |-  ( ( ( ( z  =/=  x  /\  z  =/=  y )  /\  y  =/=  x )  /\  (
( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) )  <->  ( ( z  =/=  x  /\  z  =/=  y )  /\  (
y  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
10 anass 649 . . . . . . . . . . . . . 14  |-  ( ( ( y  =/=  x  /\  y  =/=  z
)  /\  z  =/=  x )  <->  ( y  =/=  x  /\  ( y  =/=  z  /\  z  =/=  x ) ) )
11 ancom 450 . . . . . . . . . . . . . 14  |-  ( ( y  =/=  x  /\  ( y  =/=  z  /\  z  =/=  x
) )  <->  ( (
y  =/=  z  /\  z  =/=  x )  /\  y  =/=  x ) )
12 necom 2736 . . . . . . . . . . . . . . . 16  |-  ( y  =/=  z  <->  z  =/=  y )
1312anbi2ci 696 . . . . . . . . . . . . . . 15  |-  ( ( y  =/=  z  /\  z  =/=  x )  <->  ( z  =/=  x  /\  z  =/=  y ) )
1413anbi1i 695 . . . . . . . . . . . . . 14  |-  ( ( ( y  =/=  z  /\  z  =/=  x
)  /\  y  =/=  x )  <->  ( (
z  =/=  x  /\  z  =/=  y )  /\  y  =/=  x ) )
1510, 11, 143bitri 271 . . . . . . . . . . . . 13  |-  ( ( ( y  =/=  x  /\  y  =/=  z
)  /\  z  =/=  x )  <->  ( (
z  =/=  x  /\  z  =/=  y )  /\  y  =/=  x ) )
1615anbi1i 695 . . . . . . . . . . . 12  |-  ( ( ( ( y  =/=  x  /\  y  =/=  z )  /\  z  =/=  x )  /\  (
( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) )  <->  ( ( ( z  =/=  x  /\  z  =/=  y )  /\  y  =/=  x )  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
17 df-3an 975 . . . . . . . . . . . 12  |-  ( ( z  =/=  x  /\  z  =/=  y  /\  (
y  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )  <->  ( (
z  =/=  x  /\  z  =/=  y )  /\  ( y  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
189, 16, 173bitr4i 277 . . . . . . . . . . 11  |-  ( ( ( ( y  =/=  x  /\  y  =/=  z )  /\  z  =/=  x )  /\  (
( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) )  <->  ( z  =/=  x  /\  z  =/=  y  /\  ( y  =/=  x  /\  (
( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) ) ) )
197, 8, 183bitr2i 273 . . . . . . . . . 10  |-  ( ( y  =/=  x  /\  y  =/=  z  /\  (
z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )  <->  ( z  =/=  x  /\  z  =/=  y  /\  ( y  =/=  x  /\  (
( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) ) ) )
2019rexbii 2969 . . . . . . . . 9  |-  ( E. z  e.  V  ( y  =/=  x  /\  y  =/=  z  /\  (
z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )  <->  E. z  e.  V  ( z  =/=  x  /\  z  =/=  y  /\  ( y  =/=  x  /\  (
( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) ) ) )
21 rexdifpr 32090 . . . . . . . . 9  |-  ( E. z  e.  ( V 
\  { x ,  y } ) ( y  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) )  <->  E. z  e.  V  ( z  =/=  x  /\  z  =/=  y  /\  ( y  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
22 r19.42v 3021 . . . . . . . . 9  |-  ( E. z  e.  ( V 
\  { x ,  y } ) ( y  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) )  <->  ( y  =/=  x  /\  E. z  e.  ( V  \  {
x ,  y } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
2320, 21, 223bitr2i 273 . . . . . . . 8  |-  ( E. z  e.  V  ( y  =/=  x  /\  y  =/=  z  /\  (
z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )  <->  ( y  =/=  x  /\  E. z  e.  ( V  \  {
x ,  y } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
2423rexbii 2969 . . . . . . 7  |-  ( E. y  e.  V  E. z  e.  V  (
y  =/=  x  /\  y  =/=  z  /\  (
z  =/=  x  /\  ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )  <->  E. y  e.  V  ( y  =/=  x  /\  E. z  e.  ( V  \  {
x ,  y } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
255, 6, 243bitri 271 . . . . . 6  |-  ( E. z  e.  V  ( z  =/=  x  /\  E. y  e.  ( V 
\  { x ,  z } ) ( ( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) ) )  <->  E. y  e.  V  ( y  =/=  x  /\  E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
26 rexdifsn 4162 . . . . . 6  |-  ( E. z  e.  ( V 
\  { x }
) E. y  e.  ( V  \  {
x ,  z } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) )  <->  E. z  e.  V  ( z  =/=  x  /\  E. y  e.  ( V  \  { x ,  z } ) ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
27 rexdifsn 4162 . . . . . 6  |-  ( E. y  e.  ( V 
\  { x }
) E. z  e.  ( V  \  {
x ,  y } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) )  <->  E. y  e.  V  ( y  =/=  x  /\  E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
2825, 26, 273bitr4i 277 . . . . 5  |-  ( E. z  e.  ( V 
\  { x }
) E. y  e.  ( V  \  {
x ,  z } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) )  <->  E. y  e.  ( V  \  { x }
) E. z  e.  ( V  \  {
x ,  y } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) )
2928a1i 11 . . . 4  |-  ( ( V USGrph  E  /\  x  e.  V )  ->  ( E. z  e.  ( V  \  { x }
) E. y  e.  ( V  \  {
x ,  z } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) )  <->  E. y  e.  ( V  \  { x }
) E. z  e.  ( V  \  {
x ,  y } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
3029rexbidva 2975 . . 3  |-  ( V USGrph  E  ->  ( E. x  e.  V  E. z  e.  ( V  \  {
x } ) E. y  e.  ( V 
\  { x ,  z } ) ( ( ( P ` 
0 )  =  x  /\  ( P ` 
1 )  =  z  /\  ( P ` 
2 )  =  y )  /\  ( ( E `  ( F `
 0 ) )  =  { x ,  z }  /\  ( E `  ( F `  1 ) )  =  { z ,  y } ) )  <->  E. x  e.  V  E. y  e.  ( V  \  { x }
) E. z  e.  ( V  \  {
x ,  y } ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  ( ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) )
31303anbi3d 1305 . 2  |-  ( V USGrph  E  ->  ( ( F : ( 0..^ 2 ) -1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. z  e.  ( V  \  { x } ) E. y  e.  ( V  \  { x ,  z } ) ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) )  <->  ( F :
( 0..^ 2 )
-1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
321, 31bitrd 253 1  |-  ( V USGrph  E  ->  ( ( F ( V Paths  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
( 0..^ 2 )
-1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `
 0 )  =  x  /\  ( P `
 1 )  =  z  /\  ( P `
 2 )  =  y )  /\  (
( E `  ( F `  0 )
)  =  { x ,  z }  /\  ( E `  ( F `
 1 ) )  =  { z ,  y } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818    \ cdif 3478   {csn 4033   {cpr 4035   class class class wbr 4453   dom cdm 5005   -1-1->wf1 5591   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505   2c2 10597   ...cfz 11684  ..^cfzo 11804   #chash 12385   USGrph cusg 24153   Paths cpath 24323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-usgra 24156  df-wlk 24331  df-trail 24332  df-pth 24333  df-spth 24334
This theorem is referenced by: (None)
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