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Theorem 2pthon 21604
Description: A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
Assertion
Ref Expression
2pthon  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )

Proof of Theorem 2pthon
StepHypRef Expression
1 simp2 959 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  ->  A  =/=  C )
2 vex 2961 . . . . . . . . . 10  |-  i  e. 
_V
3 vex 2961 . . . . . . . . . 10  |-  j  e. 
_V
42, 3pm3.2i 443 . . . . . . . . 9  |-  ( i  e.  _V  /\  j  e.  _V )
5 eqid 2438 . . . . . . . . 9  |-  { <. 0 ,  i >. , 
<. 1 ,  j
>. }  =  { <. 0 ,  i >. , 
<. 1 ,  j
>. }
6 eqid 2438 . . . . . . . . 9  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  =  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }
74, 5, 6constr2trl 21601 . . . . . . . 8  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( V Trails  E ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
873adant3 978 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C )  ->  (
( i  =/=  j  /\  ( E `  i
)  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } )  ->  { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( V Trails  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
91, 8syl3an3 1220 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( V Trails  E ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
109imp 420 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( V Trails  E ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
11 trliswlk 21541 . . . . 5  |-  ( {
<. 0 ,  i
>. ,  <. 1 ,  j >. }  ( V Trails  E ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( V Walks  E ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } )
1210, 11syl 16 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( V Walks 
E ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
13 c0ex 9087 . . . . . . . . 9  |-  0  e.  _V
1413jctl 527 . . . . . . . 8  |-  ( A  e.  V  ->  (
0  e.  _V  /\  A  e.  V )
)
15143ad2ant1 979 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( 0  e.  _V  /\  A  e.  V ) )
16 ax-1ne0 9061 . . . . . . . . 9  |-  1  =/=  0
1716necomi 2688 . . . . . . . 8  |-  0  =/=  1
18 2ne0 10085 . . . . . . . . 9  |-  2  =/=  0
1918necomi 2688 . . . . . . . 8  |-  0  =/=  2
2017, 19pm3.2i 443 . . . . . . 7  |-  ( 0  =/=  1  /\  0  =/=  2 )
21 fvtp1g 5944 . . . . . . 7  |-  ( ( ( 0  e.  _V  /\  A  e.  V )  /\  ( 0  =/=  1  /\  0  =/=  2 ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0
)  =  A )
2215, 20, 21sylancl 645 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0
)  =  A )
23223ad2ant2 980 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0
)  =  A )
2423adantr 453 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0
)  =  A )
2516nesymi 2643 . . . . . . . 8  |-  -.  0  =  1
2613, 2opth1 4436 . . . . . . . . 9  |-  ( <.
0 ,  i >.  =  <. 1 ,  j
>.  ->  0  =  1 )
2726necon3bi 2647 . . . . . . . 8  |-  ( -.  0  =  1  ->  <. 0 ,  i >.  =/=  <. 1 ,  j
>. )
2825, 27ax-mp 8 . . . . . . 7  |-  <. 0 ,  i >.  =/=  <. 1 ,  j >.
29 opex 4429 . . . . . . . 8  |-  <. 0 ,  i >.  e.  _V
30 opex 4429 . . . . . . . 8  |-  <. 1 ,  j >.  e.  _V
31 hashprg 11668 . . . . . . . 8  |-  ( (
<. 0 ,  i
>.  e.  _V  /\  <. 1 ,  j >.  e. 
_V )  ->  ( <. 0 ,  i >.  =/=  <. 1 ,  j
>. 
<->  ( # `  { <. 0 ,  i >. ,  <. 1 ,  j
>. } )  =  2 ) )
3229, 30, 31mp2an 655 . . . . . . 7  |-  ( <.
0 ,  i >.  =/=  <. 1 ,  j
>. 
<->  ( # `  { <. 0 ,  i >. ,  <. 1 ,  j
>. } )  =  2 )
3328, 32mpbi 201 . . . . . 6  |-  ( # `  { <. 0 ,  i
>. ,  <. 1 ,  j >. } )  =  2
3433fveq2i 5733 . . . . 5  |-  ( {
<. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  ( # `  { <. 0 ,  i
>. ,  <. 1 ,  j >. } ) )  =  ( { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } `  2 )
35 2z 10314 . . . . . . . . . 10  |-  2  e.  ZZ
3635jctl 527 . . . . . . . . 9  |-  ( C  e.  V  ->  (
2  e.  ZZ  /\  C  e.  V )
)
37363ad2ant3 981 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( 2  e.  ZZ  /\  C  e.  V ) )
38 1ne2 10189 . . . . . . . . 9  |-  1  =/=  2
3919, 38pm3.2i 443 . . . . . . . 8  |-  ( 0  =/=  2  /\  1  =/=  2 )
40 fvtp3g 5946 . . . . . . . 8  |-  ( ( ( 2  e.  ZZ  /\  C  e.  V )  /\  ( 0  =/=  2  /\  1  =/=  2 ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  2
)  =  C )
4137, 39, 40sylancl 645 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  2
)  =  C )
42413ad2ant2 980 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  2
)  =  C )
4342adantr 453 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  2
)  =  C )
4434, 43syl5eq 2482 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  ( # `
 { <. 0 ,  i >. ,  <. 1 ,  j >. } ) )  =  C )
45 simpl1 961 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( V  e.  X  /\  E  e.  Y
) )
46 prex 4408 . . . . . . . 8  |-  { <. 0 ,  i >. , 
<. 1 ,  j
>. }  e.  _V
47 tpex 4710 . . . . . . . 8  |-  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V
4846, 47pm3.2i 443 . . . . . . 7  |-  ( {
<. 0 ,  i
>. ,  <. 1 ,  j >. }  e.  _V  /\ 
{ <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )
4948a1i 11 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  i >. ,  <. 1 ,  j >. }  e.  _V  /\  { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )
)
50 3simpb 956 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( A  e.  V  /\  C  e.  V
) )
51503ad2ant2 980 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( A  e.  V  /\  C  e.  V
) )
5251adantr 453 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( A  e.  V  /\  C  e.  V
) )
5345, 49, 523jca 1135 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( ( V  e.  X  /\  E  e.  Y )  /\  ( { <. 0 ,  i
>. ,  <. 1 ,  j >. }  e.  _V  /\ 
{ <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) ) )
54 iswlkon 21533 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( { <. 0 ,  i >. ,  <. 1 ,  j
>. }  e.  _V  /\  {
<. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  <->  ( { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( V Walks  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  /\  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0 )  =  A  /\  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  ( # `  { <. 0 ,  i
>. ,  <. 1 ,  j >. } ) )  =  C ) ) )
5553, 54syl 16 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  <->  ( { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( V Walks  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  /\  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  0 )  =  A  /\  ( { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } `  ( # `  { <. 0 ,  i
>. ,  <. 1 ,  j >. } ) )  =  C ) ) )
5612, 24, 44, 55mpbir3and 1138 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
574, 5, 6constr2pth 21603 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( V Paths  E ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
5857imp 420 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( V Paths 
E ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
59 ispthon 21578 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( { <. 0 ,  i >. ,  <. 1 ,  j
>. }  e.  _V  /\  {
<. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. }  <->  ( { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  /\  { <. 0 ,  i >. ,  <. 1 ,  j
>. }  ( V Paths  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) )
6053, 59syl 16 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  -> 
( { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  <->  ( { <. 0 ,  i >. , 
<. 1 ,  j
>. }  ( A ( V WalkOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }  /\  { <. 0 ,  i >. ,  <. 1 ,  j
>. }  ( V Paths  E
) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) ) )
6156, 58, 60mpbir2and 890 . 2  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )  /\  (
i  =/=  j  /\  ( E `  i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C } ) )  ->  { <. 0 ,  i
>. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. , 
<. 1 ,  B >. ,  <. 2 ,  C >. } )
6261ex 425 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/= 
C ) )  -> 
( ( i  =/=  j  /\  ( E `
 i )  =  { A ,  B }  /\  ( E `  j )  =  { B ,  C }
)  ->  { <. 0 ,  i >. ,  <. 1 ,  j >. }  ( A ( V PathOn  E ) C ) { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   {cpr 3817   {ctp 3818   <.cop 3819   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993   2c2 10051   ZZcz 10284   #chash 11620   Walks cwalk 21508   Trails ctrail 21509   Paths cpath 21510   WalkOn cwlkon 21512   PathOn cpthon 21514
This theorem is referenced by:  2pthoncl  21605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-wlk 21518  df-trail 21519  df-pth 21520  df-wlkon 21524  df-pthon 21526
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