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Theorem wlkn0 30204
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 23362 . 2  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
2 lencl 12245 . . 3  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
3 fdm 5560 . . . . . 6  |-  ( P : ( 0 ... ( # `  F
) ) --> V  ->  dom  P  =  ( 0 ... ( # `  F
) ) )
4 elnn0uz 10894 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  <->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
5 fzn0 11460 . . . . . . . 8  |-  ( ( 0 ... ( # `  F ) )  =/=  (/) 
<->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
6 neeq1 2614 . . . . . . . . . 10  |-  ( ( 0 ... ( # `  F ) )  =  dom  P  ->  (
( 0 ... ( # `
 F ) )  =/=  (/)  <->  dom  P  =/=  (/) ) )
76eqcoms 2444 . . . . . . . . 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 5559 . . . . . 6  |-  ( P : ( 0 ... ( # `  F
) ) --> V  ->  Rel  P )
14 reldm0 5053 . . . . . . 7  |-  ( Rel 
P  ->  ( P  =  (/)  <->  dom  P  =  (/) ) )
1514necon3bid 2641 . . . . . 6  |-  ( Rel 
P  ->  ( P  =/=  (/)  <->  dom  P  =/=  (/) ) )
1613, 15syl 16 . . . . 5  |-  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( P  =/=  (/)  <->  dom  P  =/=  (/) ) )
1716adantl 463 . . . 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 468 . 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 1364    e. wcel 1761    =/= wne 2604   (/)c0 3634   class class class wbr 4289   dom cdm 4836   Rel wrel 4841   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278   NN0cn0 10575   ZZ>=cuz 10857   ...cfz 11433   #chash 12099  Word cword 12217   Walks cwalk 23340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-wlk 23350
This theorem is referenced by:  wlkcpr  30215  wlk0  30217  wlkiswwlk1  30249  vfwlkniswwlkn  30265
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