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Theorem usg2wlkonot 28414
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 21376 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2wlkonotot 28404 . . . . . 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 426 . . . . 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 977 . . . 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 421 . . 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 459 . 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 2961 . . . . . . . . . . . 12  |-  f  e. 
_V
8 vex 2961 . . . . . . . . . . . 12  |-  p  e. 
_V
97, 8pm3.2i 443 . . . . . . . . . . 11  |-  ( f  e.  _V  /\  p  e.  _V )
10 is2wlk 21570 . . . . . . . . . . 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 645 . . . . . . . . . 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 3887 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 ) )  ->  { A ,  B }  =  {
( p `  0
) ,  ( p `
 1 ) } )
13123adant3 978 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  { A ,  B }  =  {
( p `  0
) ,  ( p `
 1 ) } )
1413eqeq2d 2449 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( E `
 ( f ` 
0 ) )  =  { A ,  B } 
<->  ( E `  (
f `  0 )
)  =  { ( p `  0 ) ,  ( p ` 
1 ) } ) )
15 preq12 3887 . . . . . . . . . . . . . . . . 17  |-  ( ( B  =  ( p `
 1 )  /\  C  =  ( p `  2 ) )  ->  { B ,  C }  =  {
( p `  1
) ,  ( p `
 2 ) } )
16153adant1 976 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  { B ,  C }  =  {
( p `  1
) ,  ( p `
 2 ) } )
1716eqeq2d 2449 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( E `
 ( f ` 
1 ) )  =  { B ,  C } 
<->  ( E `  (
f `  1 )
)  =  { ( p `  1 ) ,  ( p ` 
2 ) } ) )
1814, 17anbi12d 693 . . . . . . . . . . . . . 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 194 . . . . . . . . . . . . 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 1261 . . . . . . . . . . . 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 21383 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  Fun  E )
22 c0ex 9090 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  0  e.  _V
2322prid1 3914 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  { 0 ,  1 }
24 fzo0to2pr 11189 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0..^ 2 )  =  {
0 ,  1 }
2523, 24eleqtrri 2511 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  ( 0..^ 2 )
26 ffvelrn 5871 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  0  e.  (
0..^ 2 ) )  ->  ( f ` 
0 )  e.  dom  E )
2725, 26mpan2 654 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  0 )  e.  dom  E )
28 fvelrn 5869 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Fun  E  /\  (
f `  0 )  e.  dom  E )  -> 
( E `  (
f `  0 )
)  e.  ran  E
)
2927, 28sylan2 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  0 )
)  e.  ran  E
)
30 eleq1 2498 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( f `
 0 ) )  =  { A ,  B }  ->  ( ( E `  ( f `
 0 ) )  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
3129, 30syl5ibcom 213 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  0
) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
32 1ex 9091 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  1  e.  _V
3332prid2 3915 . . . . . . . . . . . . . . . . . . . . . . 23  |-  1  e.  { 0 ,  1 }
3433, 24eleqtrri 2511 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  ( 0..^ 2 )
35 ffvelrn 5871 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  1  e.  (
0..^ 2 ) )  ->  ( f ` 
1 )  e.  dom  E )
3634, 35mpan2 654 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  1 )  e.  dom  E )
37 fvelrn 5869 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Fun  E  /\  (
f `  1 )  e.  dom  E )  -> 
( E `  (
f `  1 )
)  e.  ran  E
)
3836, 37sylan2 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  1 )
)  e.  ran  E
)
39 eleq1 2498 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( f `
 1 ) )  =  { B ,  C }  ->  ( ( E `  ( f `
 1 ) )  e.  ran  E  <->  { B ,  C }  e.  ran  E ) )
4038, 39syl5ibcom 213 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  1
) )  =  { B ,  C }  ->  { B ,  C }  e.  ran  E ) )
4131, 40anim12d 548 . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . 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 24 . . . . . . . . . . . . . . . 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 426 . . . . . . . . . . . . . . 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 84 . . . . . . . . . . . . . 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 24 . . . . . . . . . . . . 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 1148 . . . . . . . . . . . 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 221 . . . . . . . . . . 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 85 . . . . . . . . . 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 208 . . . . . . . . 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 85 . . . . . . . 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 1380 . . . . . . 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 1148 . . . . . 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 77 . . . . 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 420 . . . 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 1648 . . 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 28356 . . . . 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 1157 . . . 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 453 . . 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 185 . 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 246 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 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958   {cpr 3817   <.cotp 3820   class class class wbr 4215   dom cdm 4881   ran crn 4882   Fun wfun 5451   -->wf 5453   ` cfv 5457  (class class class)co 6084   0cc0 8995   1c1 8996   2c2 10054   ...cfz 11048  ..^cfzo 11140   #chash 11623   USGrph cusg 21370   Walks cwalk 21511   2WalksOnOt c2wlkonot 28386
This theorem is referenced by:  usg2spthonot  28419  usg2spthonot0  28420  frg2woteu  28517  frg2woteqm  28521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-hash 11624  df-word 11728  df-usgra 21372  df-wlk 21521  df-wlkon 21527  df-2wlkonot 28389
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