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Theorem wlkn0 24318
Description: The set of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.)
Assertion
Ref Expression
wlkn0  |-  ( F ( V Walks  E ) P  ->  P  =/=  (/) )

Proof of Theorem wlkn0
StepHypRef Expression
1 2mwlk 24312 . 2  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
2 lencl 12538 . . 3  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
3 fdm 5740 . . . . . 6  |-  ( P : ( 0 ... ( # `  F
) ) --> V  ->  dom  P  =  ( 0 ... ( # `  F
) ) )
4 elnn0uz 11129 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  <->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
5 fzn0 11710 . . . . . . . 8  |-  ( ( 0 ... ( # `  F ) )  =/=  (/) 
<->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
6 neeq1 2748 . . . . . . . . . 10  |-  ( ( 0 ... ( # `  F ) )  =  dom  P  ->  (
( 0 ... ( # `
 F ) )  =/=  (/)  <->  dom  P  =/=  (/) ) )
76eqcoms 2479 . . . . . . . . 9  |-  ( dom 
P  =  ( 0 ... ( # `  F
) )  ->  (
( 0 ... ( # `
 F ) )  =/=  (/)  <->  dom  P  =/=  (/) ) )
87biimpcd 224 . . . . . . . 8  |-  ( ( 0 ... ( # `  F ) )  =/=  (/)  ->  ( dom  P  =  ( 0 ... ( # `  F
) )  ->  dom  P  =/=  (/) ) )
95, 8sylbir 213 . . . . . . 7  |-  ( (
# `  F )  e.  ( ZZ>= `  0 )  ->  ( dom  P  =  ( 0 ... ( # `
 F ) )  ->  dom  P  =/=  (/) ) )
104, 9sylbi 195 . . . . . 6  |-  ( (
# `  F )  e.  NN0  ->  ( dom  P  =  ( 0 ... ( # `  F
) )  ->  dom  P  =/=  (/) ) )
113, 10syl5com 30 . . . . 5  |-  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( # `  F
)  e.  NN0  ->  dom 
P  =/=  (/) ) )
1211impcom 430 . . . 4  |-  ( ( ( # `  F
)  e.  NN0  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  dom  P  =/=  (/) )
13 frel 5739 . . . . . 6  |-  ( P : ( 0 ... ( # `  F
) ) --> V  ->  Rel  P )
14 reldm0 5225 . . . . . . 7  |-  ( Rel 
P  ->  ( P  =  (/)  <->  dom  P  =  (/) ) )
1514necon3bid 2725 . . . . . 6  |-  ( Rel 
P  ->  ( P  =/=  (/)  <->  dom  P  =/=  (/) ) )
1613, 15syl 16 . . . . 5  |-  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( P  =/=  (/)  <->  dom  P  =/=  (/) ) )
1716adantl 466 . . . 4  |-  ( ( ( # `  F
)  e.  NN0  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( P  =/=  (/) 
<->  dom  P  =/=  (/) ) )
1812, 17mpbird 232 . . 3  |-  ( ( ( # `  F
)  e.  NN0  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  P  =/=  (/) )
192, 18sylan 471 . 2  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  P  =/=  (/) )
201, 19syl 16 1  |-  ( F ( V Walks  E ) P  ->  P  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   (/)c0 3790   class class class wbr 4452   dom cdm 5004   Rel wrel 5009   -->wf 5589   ` cfv 5593  (class class class)co 6294   0cc0 9502   NN0cn0 10805   ZZ>=cuz 11092   ...cfz 11682   #chash 12383  Word cword 12510   Walks cwalk 24289
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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-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 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-hash 12384  df-word 12518  df-wlk 24299
This theorem is referenced by:  wlkcpr  24320  wlkiswwlk1  24481  vfwlkniswwlkn  24497  wlk0  24552
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