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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 2734 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | wlkp 29648 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) |
3 | fdm 6745 | . . . . 5 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹))) | |
4 | 3 | eqcomd 2740 | . . . 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 29647 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
7 | elnn0uz 12920 | . . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ0 ↔ (♯‘𝐹) ∈ (ℤ≥‘0)) | |
8 | fzn0 13574 | . . . . 5 ⊢ ((0...(♯‘𝐹)) ≠ ∅ ↔ (♯‘𝐹) ∈ (ℤ≥‘0)) | |
9 | 7, 8 | sylbb2 238 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (0...(♯‘𝐹)) ≠ ∅) |
10 | 6, 9 | syl 17 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘𝐹)) ≠ ∅) |
11 | 5, 10 | eqnetrrd 3006 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → dom 𝑃 ≠ ∅) |
12 | frel 6741 | . . . 4 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Rel 𝑃) | |
13 | 2, 12 | syl 17 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → Rel 𝑃) |
14 | reldm0 5940 | . . . 4 ⊢ (Rel 𝑃 → (𝑃 = ∅ ↔ dom 𝑃 = ∅)) | |
15 | 14 | necon3bid 2982 | . . 3 ⊢ (Rel 𝑃 → (𝑃 ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
16 | 13, 15 | syl 17 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
17 | 11, 16 | mpbird 257 | 1 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∅c0 4338 class class class wbr 5147 dom cdm 5688 Rel wrel 5693 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 0cc0 11152 ℕ0cn0 12523 ℤ≥cuz 12875 ...cfz 13543 ♯chash 14365 Vtxcvtx 29027 Walkscwlks 29628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-wlks 29631 |
This theorem is referenced by: wlkvv 29659 g0wlk0 29684 wlkiswwlks1 29896 wlknewwlksn 29916 |
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