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Mirrors > Home > MPE Home > Th. List > wspn0 | Structured version Visualization version GIF version |
Description: If there are no vertices, then there are no simple paths (of any length), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 16-May-2021.) (Proof shortened by AV, 13-Mar-2022.) |
Ref | Expression |
---|---|
wspn0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
wspn0 | ⊢ (𝑉 = ∅ → (𝑁 WSPathsN 𝐺) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wspthsn 29878 | . 2 ⊢ (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} | |
2 | wwlknbp1 29874 | . . . . . 6 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1))) | |
3 | wspn0.v | . . . . . . . . . . . . 13 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | eqeq1i 2740 | . . . . . . . . . . . 12 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
5 | wrdeq 14571 | . . . . . . . . . . . 12 ⊢ ((Vtx‘𝐺) = ∅ → Word (Vtx‘𝐺) = Word ∅) | |
6 | 4, 5 | sylbi 217 | . . . . . . . . . . 11 ⊢ (𝑉 = ∅ → Word (Vtx‘𝐺) = Word ∅) |
7 | 6 | eleq2d 2825 | . . . . . . . . . 10 ⊢ (𝑉 = ∅ → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word ∅)) |
8 | 0wrd0 14575 | . . . . . . . . . 10 ⊢ (𝑤 ∈ Word ∅ ↔ 𝑤 = ∅) | |
9 | 7, 8 | bitrdi 287 | . . . . . . . . 9 ⊢ (𝑉 = ∅ → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 = ∅)) |
10 | fveq2 6907 | . . . . . . . . . . . . . . 15 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅)) | |
11 | hash0 14403 | . . . . . . . . . . . . . . 15 ⊢ (♯‘∅) = 0 | |
12 | 10, 11 | eqtrdi 2791 | . . . . . . . . . . . . . 14 ⊢ (𝑤 = ∅ → (♯‘𝑤) = 0) |
13 | 12 | eqeq1d 2737 | . . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → ((♯‘𝑤) = (𝑁 + 1) ↔ 0 = (𝑁 + 1))) |
14 | 13 | adantl 481 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑤 = ∅) → ((♯‘𝑤) = (𝑁 + 1) ↔ 0 = (𝑁 + 1))) |
15 | nn0p1gt0 12553 | . . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1)) | |
16 | 15 | gt0ne0d 11825 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ≠ 0) |
17 | eqneqall 2949 | . . . . . . . . . . . . . . 15 ⊢ ((𝑁 + 1) = 0 → ((𝑁 + 1) ≠ 0 → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) | |
18 | 17 | eqcoms 2743 | . . . . . . . . . . . . . 14 ⊢ (0 = (𝑁 + 1) → ((𝑁 + 1) ≠ 0 → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) |
19 | 16, 18 | syl5com 31 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → (0 = (𝑁 + 1) → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) |
20 | 19 | adantr 480 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑤 = ∅) → (0 = (𝑁 + 1) → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) |
21 | 14, 20 | sylbid 240 | . . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑤 = ∅) → ((♯‘𝑤) = (𝑁 + 1) → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) |
22 | 21 | expcom 413 | . . . . . . . . . 10 ⊢ (𝑤 = ∅ → (𝑁 ∈ ℕ0 → ((♯‘𝑤) = (𝑁 + 1) → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))) |
23 | 22 | com23 86 | . . . . . . . . 9 ⊢ (𝑤 = ∅ → ((♯‘𝑤) = (𝑁 + 1) → (𝑁 ∈ ℕ0 → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))) |
24 | 9, 23 | biimtrdi 253 | . . . . . . . 8 ⊢ (𝑉 = ∅ → (𝑤 ∈ Word (Vtx‘𝐺) → ((♯‘𝑤) = (𝑁 + 1) → (𝑁 ∈ ℕ0 → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)))) |
25 | 24 | com14 96 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑤 ∈ Word (Vtx‘𝐺) → ((♯‘𝑤) = (𝑁 + 1) → (𝑉 = ∅ → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)))) |
26 | 25 | 3imp 1110 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1)) → (𝑉 = ∅ → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) |
27 | 2, 26 | syl 17 | . . . . 5 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑉 = ∅ → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) |
28 | 27 | impcom 407 | . . . 4 ⊢ ((𝑉 = ∅ ∧ 𝑤 ∈ (𝑁 WWalksN 𝐺)) → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤) |
29 | 28 | ralrimiva 3144 | . . 3 ⊢ (𝑉 = ∅ → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤) |
30 | rabeq0 4394 | . . 3 ⊢ ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅ ↔ ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤) | |
31 | 29, 30 | sylibr 234 | . 2 ⊢ (𝑉 = ∅ → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} = ∅) |
32 | 1, 31 | eqtrid 2787 | 1 ⊢ (𝑉 = ∅ → (𝑁 WSPathsN 𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {crab 3433 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 ℕ0cn0 12524 ♯chash 14366 Word cword 14549 Vtxcvtx 29028 SPathscspths 29746 WWalksN cwwlksn 29856 WSPathsN cwwspthsn 29858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-wwlks 29860 df-wwlksn 29861 df-wspthsn 29863 |
This theorem is referenced by: (None) |
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