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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 2725 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | wlkp 29502 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) |
3 | fdm 6732 | . . . . 5 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹))) | |
4 | 3 | eqcomd 2731 | . . . 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 29501 | . . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
7 | elnn0uz 12900 | . . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ0 ↔ (♯‘𝐹) ∈ (ℤ≥‘0)) | |
8 | fzn0 13550 | . . . . 5 ⊢ ((0...(♯‘𝐹)) ≠ ∅ ↔ (♯‘𝐹) ∈ (ℤ≥‘0)) | |
9 | 7, 8 | sylbb2 237 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (0...(♯‘𝐹)) ≠ ∅) |
10 | 6, 9 | syl 17 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘𝐹)) ≠ ∅) |
11 | 5, 10 | eqnetrrd 2998 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → dom 𝑃 ≠ ∅) |
12 | frel 6728 | . . . 4 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Rel 𝑃) | |
13 | 2, 12 | syl 17 | . . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → Rel 𝑃) |
14 | reldm0 5930 | . . . 4 ⊢ (Rel 𝑃 → (𝑃 = ∅ ↔ dom 𝑃 = ∅)) | |
15 | 14 | necon3bid 2974 | . . 3 ⊢ (Rel 𝑃 → (𝑃 ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
16 | 13, 15 | syl 17 | . 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
17 | 11, 16 | mpbird 256 | 1 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∅c0 4322 class class class wbr 5149 dom cdm 5678 Rel wrel 5683 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 0cc0 11140 ℕ0cn0 12505 ℤ≥cuz 12855 ...cfz 13519 ♯chash 14325 Vtxcvtx 28881 Walkscwlks 29482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-wlks 29485 |
This theorem is referenced by: wlkvv 29513 g0wlk0 29538 wlkiswwlks1 29750 wlknewwlksn 29770 |
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