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Theorem usg2wlkonot 28080
Description: A walk of length 2 between two vertices as ordered triple in an undirected simple graph. This theorem would also hold for undirected multigraphs, but to proof this the cases  A  =  B and/or  B  =  C must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
Assertion
Ref Expression
usg2wlkonot  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <-> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )

Proof of Theorem usg2wlkonot
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 21324 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2wlkonotot 28070 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
32expcom 425 . . . . 5  |-  ( ( A  e.  V  /\  C  e.  V )  ->  ( ( V  e. 
_V  /\  E  e.  _V )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
433adant2 976 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( V  e. 
_V  /\  E  e.  _V )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
54impcom 420 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
61, 5sylan 458 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
7 vex 2919 . . . . . . . . . . . 12  |-  f  e. 
_V
8 vex 2919 . . . . . . . . . . . 12  |-  p  e. 
_V
97, 8pm3.2i 442 . . . . . . . . . . 11  |-  ( f  e.  _V  /\  p  e.  _V )
10 is2wlk 21518 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( f  e.  _V  /\  p  e.  _V )
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) ) ) )
111, 9, 10sylancl 644 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) ) ) )
12 preq12 3845 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 ) )  ->  { A ,  B }  =  {
( p `  0
) ,  ( p `
 1 ) } )
13123adant3 977 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  { A ,  B }  =  {
( p `  0
) ,  ( p `
 1 ) } )
1413eqeq2d 2415 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( E `
 ( f ` 
0 ) )  =  { A ,  B } 
<->  ( E `  (
f `  0 )
)  =  { ( p `  0 ) ,  ( p ` 
1 ) } ) )
15 preq12 3845 . . . . . . . . . . . . . . . . 17  |-  ( ( B  =  ( p `
 1 )  /\  C  =  ( p `  2 ) )  ->  { B ,  C }  =  {
( p `  1
) ,  ( p `
 2 ) } )
16153adant1 975 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  { B ,  C }  =  {
( p `  1
) ,  ( p `
 2 ) } )
1716eqeq2d 2415 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( E `
 ( f ` 
1 ) )  =  { B ,  C } 
<->  ( E `  (
f `  1 )
)  =  { ( p `  1 ) ,  ( p ` 
2 ) } ) )
1814, 17anbi12d 692 . . . . . . . . . . . . . 14  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( ( E `  ( f `
 0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1 ) )  =  { B ,  C } )  <->  ( ( E `  ( f `  0 ) )  =  { ( p `
 0 ) ,  ( p `  1
) }  /\  ( E `  ( f `  1 ) )  =  { ( p `
 1 ) ,  ( p `  2
) } ) ) )
1918bicomd 193 . . . . . . . . . . . . 13  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( ( E `  ( f `
 0 ) )  =  { ( p `
 0 ) ,  ( p `  1
) }  /\  ( E `  ( f `  1 ) )  =  { ( p `
 1 ) ,  ( p `  2
) } )  <->  ( ( E `  ( f `  0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1 ) )  =  { B ,  C } ) ) )
20193anbi3d 1260 . . . . . . . . . . . 12  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  { A ,  B }  /\  ( E `  (
f `  1 )
)  =  { B ,  C } ) ) ) )
21 usgrafun 21331 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  Fun  E )
22 c0ex 9041 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  0  e.  _V
2322prid1 3872 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  { 0 ,  1 }
24 fzo0to2pr 11139 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0..^ 2 )  =  {
0 ,  1 }
2523, 24eleqtrri 2477 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  ( 0..^ 2 )
26 ffvelrn 5827 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  0  e.  (
0..^ 2 ) )  ->  ( f ` 
0 )  e.  dom  E )
2725, 26mpan2 653 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  0 )  e.  dom  E )
28 fvelrn 5825 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Fun  E  /\  (
f `  0 )  e.  dom  E )  -> 
( E `  (
f `  0 )
)  e.  ran  E
)
2927, 28sylan2 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  0 )
)  e.  ran  E
)
30 eleq1 2464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( f `
 0 ) )  =  { A ,  B }  ->  ( ( E `  ( f `
 0 ) )  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
3129, 30syl5ibcom 212 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  0
) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
32 1ex 9042 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  1  e.  _V
3332prid2 3873 . . . . . . . . . . . . . . . . . . . . . . 23  |-  1  e.  { 0 ,  1 }
3433, 24eleqtrri 2477 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  ( 0..^ 2 )
35 ffvelrn 5827 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  1  e.  (
0..^ 2 ) )  ->  ( f ` 
1 )  e.  dom  E )
3634, 35mpan2 653 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  1 )  e.  dom  E )
37 fvelrn 5825 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Fun  E  /\  (
f `  1 )  e.  dom  E )  -> 
( E `  (
f `  1 )
)  e.  ran  E
)
3836, 37sylan2 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  1 )
)  e.  ran  E
)
39 eleq1 2464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( f `
 1 ) )  =  { B ,  C }  ->  ( ( E `  ( f `
 1 ) )  e.  ran  E  <->  { B ,  C }  e.  ran  E ) )
4038, 39syl5ibcom 212 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  1
) )  =  { B ,  C }  ->  { B ,  C }  e.  ran  E ) )
4131, 40anim12d 547 . . . . . . . . . . . . . . . . . 18  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
4221, 41sylan 458 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
4342a1d 23 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  (
( ( E `  ( f `  0
) )  =  { A ,  B }  /\  ( E `  (
f `  1 )
)  =  { B ,  C } )  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
4443expcom 425 . . . . . . . . . . . . . . 15  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4544com24 83 . . . . . . . . . . . . . 14  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( ( ( E `  ( f `  0
) )  =  { A ,  B }  /\  ( E `  (
f `  1 )
)  =  { B ,  C } )  -> 
( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4645a1d 23 . . . . . . . . . . . . 13  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( p : ( 0 ... 2 ) --> V  ->  ( ( ( E `  ( f `
 0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1 ) )  =  { B ,  C } )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) ) )
47463imp 1147 . . . . . . . . . . . 12  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
) )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
4820, 47syl6bi 220 . . . . . . . . . . 11  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4948com14 84 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
5011, 49sylbid 207 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
5150com14 84 . . . . . . . 8  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
5251exp3acom3r 1376 . . . . . . 7  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) ) )
53523imp 1147 . . . . . 6  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
5453com13 76 . . . . 5  |-  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  (
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
5554imp 419 . . . 4  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
5655exlimdvv 1644 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
57 usg2wlk 28049 . . . . 5  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
58573expib 1156 . . . 4  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
5958adantr 452 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
6056, 59impbid 184 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )  <->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
616, 60bitrd 245 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <-> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916   {cpr 3775   <.cotp 3778   class class class wbr 4172   dom cdm 4837   ran crn 4838   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946   1c1 8947   2c2 10005   ...cfz 10999  ..^cfzo 11090   #chash 11573   USGrph cusg 21318   Walks cwalk 21459   2WalksOnOt c2wlkonot 28052
This theorem is referenced by:  usg2spthonot  28085  usg2spthonot0  28086  frg2woteu  28158  frg2woteqm  28162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-usgra 21320  df-wlk 21469  df-wlkon 21475  df-2wlkonot 28055
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