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Mirrors > Home > MPE Home > Th. List > wlkn0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
wlkn0 | ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → 𝑃 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2mwlk 26049 | . 2 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉)) | |
2 | lencl 13179 | . . 3 ⊢ (𝐹 ∈ Word dom 𝐸 → (#‘𝐹) ∈ ℕ0) | |
3 | fdm 5964 | . . . . . 6 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 → dom 𝑃 = (0...(#‘𝐹))) | |
4 | elnn0uz 11601 | . . . . . . 7 ⊢ ((#‘𝐹) ∈ ℕ0 ↔ (#‘𝐹) ∈ (ℤ≥‘0)) | |
5 | fzn0 12226 | . . . . . . . 8 ⊢ ((0...(#‘𝐹)) ≠ ∅ ↔ (#‘𝐹) ∈ (ℤ≥‘0)) | |
6 | neeq1 2844 | . . . . . . . . . 10 ⊢ ((0...(#‘𝐹)) = dom 𝑃 → ((0...(#‘𝐹)) ≠ ∅ ↔ dom 𝑃 ≠ ∅)) | |
7 | 6 | eqcoms 2618 | . . . . . . . . 9 ⊢ (dom 𝑃 = (0...(#‘𝐹)) → ((0...(#‘𝐹)) ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
8 | 7 | biimpcd 238 | . . . . . . . 8 ⊢ ((0...(#‘𝐹)) ≠ ∅ → (dom 𝑃 = (0...(#‘𝐹)) → dom 𝑃 ≠ ∅)) |
9 | 5, 8 | sylbir 224 | . . . . . . 7 ⊢ ((#‘𝐹) ∈ (ℤ≥‘0) → (dom 𝑃 = (0...(#‘𝐹)) → dom 𝑃 ≠ ∅)) |
10 | 4, 9 | sylbi 206 | . . . . . 6 ⊢ ((#‘𝐹) ∈ ℕ0 → (dom 𝑃 = (0...(#‘𝐹)) → dom 𝑃 ≠ ∅)) |
11 | 3, 10 | syl5com 31 | . . . . 5 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 → ((#‘𝐹) ∈ ℕ0 → dom 𝑃 ≠ ∅)) |
12 | 11 | impcom 445 | . . . 4 ⊢ (((#‘𝐹) ∈ ℕ0 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → dom 𝑃 ≠ ∅) |
13 | frel 5963 | . . . . . 6 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 → Rel 𝑃) | |
14 | reldm0 5264 | . . . . . . 7 ⊢ (Rel 𝑃 → (𝑃 = ∅ ↔ dom 𝑃 = ∅)) | |
15 | 14 | necon3bid 2826 | . . . . . 6 ⊢ (Rel 𝑃 → (𝑃 ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
16 | 13, 15 | syl 17 | . . . . 5 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 → (𝑃 ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
17 | 16 | adantl 481 | . . . 4 ⊢ (((#‘𝐹) ∈ ℕ0 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → (𝑃 ≠ ∅ ↔ dom 𝑃 ≠ ∅)) |
18 | 12, 17 | mpbird 246 | . . 3 ⊢ (((#‘𝐹) ∈ ℕ0 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → 𝑃 ≠ ∅) |
19 | 2, 18 | sylan 487 | . 2 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → 𝑃 ≠ ∅) |
20 | 1, 19 | syl 17 | 1 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → 𝑃 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 class class class wbr 4583 dom cdm 5038 Rel wrel 5043 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℕ0cn0 11169 ℤ≥cuz 11563 ...cfz 12197 #chash 12979 Word cword 13146 Walks cwalk 26026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-wlk 26036 |
This theorem is referenced by: wlkcpr 26057 wlkiswwlk1 26218 vfwlkniswwlkn 26234 wlk0 26289 |
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