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Theorem wlkn0 30284
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 23432 . 2  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
2 lencl 12254 . . 3  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
3 fdm 5568 . . . . . 6  |-  ( P : ( 0 ... ( # `  F
) ) --> V  ->  dom  P  =  ( 0 ... ( # `  F
) ) )
4 elnn0uz 10903 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  <->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
5 fzn0 11469 . . . . . . . 8  |-  ( ( 0 ... ( # `  F ) )  =/=  (/) 
<->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
6 neeq1 2621 . . . . . . . . . 10  |-  ( ( 0 ... ( # `  F ) )  =  dom  P  ->  (
( 0 ... ( # `
 F ) )  =/=  (/)  <->  dom  P  =/=  (/) ) )
76eqcoms 2446 . . . . . . . . 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 5567 . . . . . 6  |-  ( P : ( 0 ... ( # `  F
) ) --> V  ->  Rel  P )
14 reldm0 5062 . . . . . . 7  |-  ( Rel 
P  ->  ( P  =  (/)  <->  dom  P  =  (/) ) )
1514necon3bid 2648 . . . . . 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 1369    e. wcel 1756    =/= wne 2611   (/)c0 3642   class class class wbr 4297   dom cdm 4845   Rel wrel 4850   -->wf 5419   ` cfv 5423  (class class class)co 6096   0cc0 9287   NN0cn0 10584   ZZ>=cuz 10866   ...cfz 11442   #chash 12108  Word cword 12226   Walks cwalk 23410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-hash 12109  df-word 12234  df-wlk 23420
This theorem is referenced by:  wlkcpr  30295  wlk0  30297  wlkiswwlk1  30329  vfwlkniswwlkn  30345
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