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Mirrors > Home > MPE Home > Th. List > wlkn0 | Structured version Visualization version GIF version |
Description: The sequence 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.) (Revised by AV, 2-Jan-2021.) |
Ref | Expression |
---|---|
wlkn0 | ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | wlkp 26921 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) |
3 | fdm 6290 | . . . . 5 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹))) | |
4 | 3 | eqcomd 2831 | . . . 4 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (0...(♯‘𝐹)) = dom 𝑃) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘𝐹)) = dom 𝑃) |
6 | wlkcl 26920 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
7 | elnn0uz 12014 | . . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ0 ↔ (♯‘𝐹) ∈ (ℤ≥‘0)) | |
8 | fzn0 12655 | . . . . 5 ⊢ ((0...(♯‘𝐹)) ≠ ∅ ↔ (♯‘𝐹) ∈ (ℤ≥‘0)) | |
9 | 7, 8 | sylbb2 230 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (0...(♯‘𝐹)) ≠ ∅) |
10 | 6, 9 | syl 17 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘𝐹)) ≠ ∅) |
11 | 5, 10 | eqnetrrd 3067 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → dom 𝑃 ≠ ∅) |
12 | frel 6287 | . . . 4 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Rel 𝑃) | |
13 | 2, 12 | syl 17 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → Rel 𝑃) |
14 | reldm0 5579 | . . . 4 ⊢ (Rel 𝑃 → (𝑃 = ∅ ↔ dom 𝑃 = ∅)) | |
15 | 14 | necon3bid 3043 | . . 3 ⊢ (Rel 𝑃 → (𝑃 ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
16 | 13, 15 | syl 17 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
17 | 11, 16 | mpbird 249 | 1 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∅c0 4146 class class class wbr 4875 dom cdm 5346 Rel wrel 5351 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 0cc0 10259 ℕ0cn0 11625 ℤ≥cuz 11975 ...cfz 12626 ♯chash 13417 Vtxcvtx 26301 Walkscwlks 26901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ifp 1090 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-hash 13418 df-word 13582 df-wlks 26904 |
This theorem is referenced by: wlkvv 26931 g0wlk0 26956 wlkiswwlks1 27173 wlknewwlksn 27194 |
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