![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2778 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | wlkp 26968 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) |
3 | fdm 6301 | . . . . 5 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹))) | |
4 | 3 | eqcomd 2784 | . . . 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 26967 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
7 | elnn0uz 12035 | . . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ0 ↔ (♯‘𝐹) ∈ (ℤ≥‘0)) | |
8 | fzn0 12676 | . . . . 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 3037 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → dom 𝑃 ≠ ∅) |
12 | frel 6298 | . . . 4 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Rel 𝑃) | |
13 | 2, 12 | syl 17 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → Rel 𝑃) |
14 | reldm0 5590 | . . . 4 ⊢ (Rel 𝑃 → (𝑃 = ∅ ↔ dom 𝑃 = ∅)) | |
15 | 14 | necon3bid 3013 | . . 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 1601 ∈ wcel 2107 ≠ wne 2969 ∅c0 4141 class class class wbr 4888 dom cdm 5357 Rel wrel 5362 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 0cc0 10274 ℕ0cn0 11646 ℤ≥cuz 11996 ...cfz 12647 ♯chash 13439 Vtxcvtx 26348 Walkscwlks 26948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ifp 1047 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-n0 11647 df-z 11733 df-uz 11997 df-fz 12648 df-fzo 12789 df-hash 13440 df-word 13604 df-wlks 26951 |
This theorem is referenced by: wlkvv 26978 g0wlk0 27003 wlkiswwlks1 27220 wlknewwlksn 27241 |
Copyright terms: Public domain | W3C validator |