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Theorem 2spontn0vne 25089
Description: If the set of simple paths of length 2 between two vertices (in a graph) is not empty, the two vertices must be not equal. (Contributed by Alexander van der Vekens, 3-Mar-2018.)
Assertion
Ref Expression
2spontn0vne  |-  ( ( X ( V 2SPathOnOt  E ) Y )  =/=  (/)  ->  X  =/=  Y )

Proof of Theorem 2spontn0vne
Dummy variables  b 
f  i  p  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3793 . 2  |-  ( ( X ( V 2SPathOnOt  E ) Y )  =/=  (/)  <->  E. t 
t  e.  ( X ( V 2SPathOnOt  E ) Y ) )
2 2spthonot3v 25078 . . . 4  |-  ( t  e.  ( X ( V 2SPathOnOt  E ) Y )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )  /\  t  e.  (
( V  X.  V
)  X.  V ) ) )
3 el2spthonot 25072 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  (
t  e.  ( X ( V 2SPathOnOt  E ) Y )  <->  E. b  e.  V  ( t  =  <. X ,  b ,  Y >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( X  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  Y  =  ( p `  2
) ) ) ) ) )
4 vex 3109 . . . . . . . . . . . . . 14  |-  f  e. 
_V
5 vex 3109 . . . . . . . . . . . . . 14  |-  p  e. 
_V
6 isspth 24773 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( f  e.  _V  /\  p  e.  _V )
)  ->  ( f
( V SPaths  E )
p  <->  ( f ( V Trails  E ) p  /\  Fun  `' p
) ) )
7 istrl2 24742 . . . . . . . . . . . . . . . 16  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( f  e.  _V  /\  p  e.  _V )
)  ->  ( f
( V Trails  E )
p  <->  ( f : ( 0..^ ( # `  f ) ) -1-1-> dom  E  /\  p : ( 0 ... ( # `  f ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 i ) )  =  { ( p `
 i ) ,  ( p `  (
i  +  1 ) ) } ) ) )
87anbi1d 702 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( f  e.  _V  /\  p  e.  _V )
)  ->  ( (
f ( V Trails  E
) p  /\  Fun  `' p )  <->  ( (
f : ( 0..^ ( # `  f
) ) -1-1-> dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  i ) )  =  { ( p `  i ) ,  ( p `  ( i  +  1 ) ) } )  /\  Fun  `' p ) ) )
96, 8bitrd 253 . . . . . . . . . . . . . 14  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( f  e.  _V  /\  p  e.  _V )
)  ->  ( f
( V SPaths  E )
p  <->  ( ( f : ( 0..^ (
# `  f )
) -1-1-> dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  i ) )  =  { ( p `  i ) ,  ( p `  ( i  +  1 ) ) } )  /\  Fun  `' p ) ) )
104, 5, 9mpanr12 683 . . . . . . . . . . . . 13  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V SPaths  E ) p  <->  ( (
f : ( 0..^ ( # `  f
) ) -1-1-> dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  i ) )  =  { ( p `  i ) ,  ( p `  ( i  +  1 ) ) } )  /\  Fun  `' p ) ) )
1110adantr 463 . . . . . . . . . . . 12  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  (
f ( V SPaths  E
) p  <->  ( (
f : ( 0..^ ( # `  f
) ) -1-1-> dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  i ) )  =  { ( p `  i ) ,  ( p `  ( i  +  1 ) ) } )  /\  Fun  `' p ) ) )
1211adantr 463 . . . . . . . . . . 11  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )
)  /\  b  e.  V )  ->  (
f ( V SPaths  E
) p  <->  ( (
f : ( 0..^ ( # `  f
) ) -1-1-> dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  i ) )  =  { ( p `  i ) ,  ( p `  ( i  +  1 ) ) } )  /\  Fun  `' p ) ) )
13 df-f1 5575 . . . . . . . . . . . . . 14  |-  ( p : ( 0 ... ( # `  f
) ) -1-1-> V  <->  ( p : ( 0 ... ( # `  f
) ) --> V  /\  Fun  `' p ) )
14 eqidd 2455 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  f )  =  2  ->  p  =  p )
15 oveq2 6278 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  f )  =  2  ->  (
0 ... ( # `  f
) )  =  ( 0 ... 2 ) )
16 eqidd 2455 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  f )  =  2  ->  V  =  V )
1714, 15, 16f1eq123d 5793 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  f )  =  2  ->  (
p : ( 0 ... ( # `  f
) ) -1-1-> V  <->  p :
( 0 ... 2
) -1-1-> V ) )
18 eqid 2454 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0 ... 2 )  =  ( 0 ... 2
)
1918f13idfv 12088 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p : ( 0 ... 2 ) -1-1-> V  <->  ( p : ( 0 ... 2 ) --> V  /\  ( ( p ` 
0 )  =/=  (
p `  1 )  /\  ( p `  0
)  =/=  ( p `
 2 )  /\  ( p `  1
)  =/=  ( p `
 2 ) ) ) )
20 simpr2 1001 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( p : ( 0 ... 2 ) --> V  /\  ( ( p `
 0 )  =/=  ( p `  1
)  /\  ( p `  0 )  =/=  ( p `  2
)  /\  ( p `  1 )  =/=  ( p `  2
) ) )  -> 
( p `  0
)  =/=  ( p `
 2 ) )
2119, 20sylbi 195 . . . . . . . . . . . . . . . . . . . . 21  |-  ( p : ( 0 ... 2 ) -1-1-> V  -> 
( p `  0
)  =/=  ( p `
 2 ) )
2221a1d 25 . . . . . . . . . . . . . . . . . . . 20  |-  ( p : ( 0 ... 2 ) -1-1-> V  -> 
( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  /\  b  e.  V )  ->  (
p `  0 )  =/=  ( p `  2
) ) )
2317, 22syl6bi 228 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  f )  =  2  ->  (
p : ( 0 ... ( # `  f
) ) -1-1-> V  -> 
( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  /\  b  e.  V )  ->  (
p `  0 )  =/=  ( p `  2
) ) ) )
2423com3l 81 . . . . . . . . . . . . . . . . . 18  |-  ( p : ( 0 ... ( # `  f
) ) -1-1-> V  -> 
( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  /\  b  e.  V )  ->  (
( # `  f )  =  2  ->  (
p `  0 )  =/=  ( p `  2
) ) ) )
2524imp31 430 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p : ( 0 ... ( # `  f ) ) -1-1-> V  /\  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )
)  /\  b  e.  V ) )  /\  ( # `  f )  =  2 )  -> 
( p `  0
)  =/=  ( p `
 2 ) )
2625adantr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( p : ( 0 ... ( # `
 f ) )
-1-1-> V  /\  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  /\  b  e.  V ) )  /\  ( # `  f )  =  2 )  /\  ( X  =  (
p `  0 )  /\  b  =  (
p `  1 )  /\  Y  =  (
p `  2 )
) )  ->  (
p `  0 )  =/=  ( p `  2
) )
27 simpl 455 . . . . . . . . . . . . . . . . . . 19  |-  ( ( X  =  ( p `
 0 )  /\  Y  =  ( p `  2 ) )  ->  X  =  ( p `  0 ) )
28 simpr 459 . . . . . . . . . . . . . . . . . . 19  |-  ( ( X  =  ( p `
 0 )  /\  Y  =  ( p `  2 ) )  ->  Y  =  ( p `  2 ) )
2927, 28neeq12d 2733 . . . . . . . . . . . . . . . . . 18  |-  ( ( X  =  ( p `
 0 )  /\  Y  =  ( p `  2 ) )  ->  ( X  =/= 
Y  <->  ( p ` 
0 )  =/=  (
p `  2 )
) )
30293adant2 1013 . . . . . . . . . . . . . . . . 17  |-  ( ( X  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  Y  =  ( p `  2 ) )  ->  ( X  =/= 
Y  <->  ( p ` 
0 )  =/=  (
p `  2 )
) )
3130adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( p : ( 0 ... ( # `
 f ) )
-1-1-> V  /\  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  /\  b  e.  V ) )  /\  ( # `  f )  =  2 )  /\  ( X  =  (
p `  0 )  /\  b  =  (
p `  1 )  /\  Y  =  (
p `  2 )
) )  ->  ( X  =/=  Y  <->  ( p `  0 )  =/=  ( p `  2
) ) )
3226, 31mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( ( ( p : ( 0 ... ( # `
 f ) )
-1-1-> V  /\  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  /\  b  e.  V ) )  /\  ( # `  f )  =  2 )  /\  ( X  =  (
p `  0 )  /\  b  =  (
p `  1 )  /\  Y  =  (
p `  2 )
) )  ->  X  =/=  Y )
3332exp41 608 . . . . . . . . . . . . . 14  |-  ( p : ( 0 ... ( # `  f
) ) -1-1-> V  -> 
( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  /\  b  e.  V )  ->  (
( # `  f )  =  2  ->  (
( X  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  Y  =  ( p `  2 ) )  ->  X  =/=  Y ) ) ) )
3413, 33sylbir 213 . . . . . . . . . . . . 13  |-  ( ( p : ( 0 ... ( # `  f
) ) --> V  /\  Fun  `' p )  ->  (
( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )
)  /\  b  e.  V )  ->  (
( # `  f )  =  2  ->  (
( X  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  Y  =  ( p `  2 ) )  ->  X  =/=  Y ) ) ) )
35343ad2antl2 1157 . . . . . . . . . . . 12  |-  ( ( ( f : ( 0..^ ( # `  f
) ) -1-1-> dom  E  /\  p : ( 0 ... ( # `  f
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  i ) )  =  { ( p `  i ) ,  ( p `  ( i  +  1 ) ) } )  /\  Fun  `' p )  ->  (
( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )
)  /\  b  e.  V )  ->  (
( # `  f )  =  2  ->  (
( X  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  Y  =  ( p `  2 ) )  ->  X  =/=  Y ) ) ) )
3635com12 31 . . . . . . . . . . 11  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )
)  /\  b  e.  V )  ->  (
( ( f : ( 0..^ ( # `  f ) ) -1-1-> dom  E  /\  p : ( 0 ... ( # `  f ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 i ) )  =  { ( p `
 i ) ,  ( p `  (
i  +  1 ) ) } )  /\  Fun  `' p )  ->  (
( # `  f )  =  2  ->  (
( X  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  Y  =  ( p `  2 ) )  ->  X  =/=  Y ) ) ) )
3712, 36sylbid 215 . . . . . . . . . 10  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )
)  /\  b  e.  V )  ->  (
f ( V SPaths  E
) p  ->  (
( # `  f )  =  2  ->  (
( X  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  Y  =  ( p `  2 ) )  ->  X  =/=  Y ) ) ) )
38373impd 1208 . . . . . . . . 9  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )
)  /\  b  e.  V )  ->  (
( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( X  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  Y  =  ( p `  2 ) ) )  ->  X  =/=  Y ) )
3938exlimdvv 1730 . . . . . . . 8  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )
)  /\  b  e.  V )  ->  ( E. f E. p ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( X  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  Y  =  ( p ` 
2 ) ) )  ->  X  =/=  Y
) )
4039adantld 465 . . . . . . 7  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V )
)  /\  b  e.  V )  ->  (
( t  =  <. X ,  b ,  Y >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( X  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  Y  =  ( p `  2 ) ) ) )  ->  X  =/=  Y ) )
4140rexlimdva 2946 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  ( E. b  e.  V  ( t  =  <. X ,  b ,  Y >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( X  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  Y  =  ( p `  2 ) ) ) )  ->  X  =/=  Y ) )
423, 41sylbid 215 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  (
t  e.  ( X ( V 2SPathOnOt  E ) Y )  ->  X  =/=  Y ) )
43423adant3 1014 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( X  e.  V  /\  Y  e.  V
)  /\  t  e.  ( ( V  X.  V )  X.  V
) )  ->  (
t  e.  ( X ( V 2SPathOnOt  E ) Y )  ->  X  =/=  Y ) )
442, 43mpcom 36 . . 3  |-  ( t  e.  ( X ( V 2SPathOnOt  E ) Y )  ->  X  =/=  Y
)
4544exlimiv 1727 . 2  |-  ( E. t  t  e.  ( X ( V 2SPathOnOt  E ) Y )  ->  X  =/=  Y )
461, 45sylbi 195 1  |-  ( ( X ( V 2SPathOnOt  E ) Y )  =/=  (/)  ->  X  =/=  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106   (/)c0 3783   {cpr 4018   <.cotp 4024   class class class wbr 4439    X. cxp 4986   `'ccnv 4987   dom cdm 4988   Fun wfun 5564   -->wf 5566   -1-1->wf1 5567   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484   2c2 10581   ...cfz 11675  ..^cfzo 11799   #chash 12387   Trails ctrail 24701   SPaths cspath 24703   2SPathOnOt c2pthonot 25059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-wlk 24710  df-trail 24711  df-pth 24712  df-spth 24713  df-wlkon 24716  df-spthon 24719  df-2spthonot 25062
This theorem is referenced by:  usg2spthonot  25090  usg2spthonot0  25091  2spot2iun2spont  25093
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