|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 29868 | . 2 ⊢ (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤} | |
| 2 | wwlknbp1 29864 | . . . . . 6 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (𝑁 + 1))) | |
| 3 | wspn0.v | . . . . . . . . . . . . 13 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 4 | 3 | eqeq1i 2742 | . . . . . . . . . . . 12 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) | 
| 5 | wrdeq 14574 | . . . . . . . . . . . 12 ⊢ ((Vtx‘𝐺) = ∅ → Word (Vtx‘𝐺) = Word ∅) | |
| 6 | 4, 5 | sylbi 217 | . . . . . . . . . . 11 ⊢ (𝑉 = ∅ → Word (Vtx‘𝐺) = Word ∅) | 
| 7 | 6 | eleq2d 2827 | . . . . . . . . . 10 ⊢ (𝑉 = ∅ → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word ∅)) | 
| 8 | 0wrd0 14578 | . . . . . . . . . 10 ⊢ (𝑤 ∈ Word ∅ ↔ 𝑤 = ∅) | |
| 9 | 7, 8 | bitrdi 287 | . . . . . . . . 9 ⊢ (𝑉 = ∅ → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 = ∅)) | 
| 10 | fveq2 6906 | . . . . . . . . . . . . . . 15 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅)) | |
| 11 | hash0 14406 | . . . . . . . . . . . . . . 15 ⊢ (♯‘∅) = 0 | |
| 12 | 10, 11 | eqtrdi 2793 | . . . . . . . . . . . . . 14 ⊢ (𝑤 = ∅ → (♯‘𝑤) = 0) | 
| 13 | 12 | eqeq1d 2739 | . . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → ((♯‘𝑤) = (𝑁 + 1) ↔ 0 = (𝑁 + 1))) | 
| 14 | 13 | adantl 481 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑤 = ∅) → ((♯‘𝑤) = (𝑁 + 1) ↔ 0 = (𝑁 + 1))) | 
| 15 | nn0p1gt0 12555 | . . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1)) | |
| 16 | 15 | gt0ne0d 11827 | . . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ≠ 0) | 
| 17 | eqneqall 2951 | . . . . . . . . . . . . . . 15 ⊢ ((𝑁 + 1) = 0 → ((𝑁 + 1) ≠ 0 → ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) | |
| 18 | 17 | eqcoms 2745 | . . . . . . . . . . . . . 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 1111 | . . . . . 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 3146 | . . 3 ⊢ (𝑉 = ∅ → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ∃𝑓 𝑓(SPaths‘𝐺)𝑤) | 
| 30 | rabeq0 4388 | . . 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 2789 | 1 ⊢ (𝑉 = ∅ → (𝑁 WSPathsN 𝐺) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ℕ0cn0 12526 ♯chash 14369 Word cword 14552 Vtxcvtx 29013 SPathscspths 29731 WWalksN cwwlksn 29846 WSPathsN cwwspthsn 29848 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-wwlks 29850 df-wwlksn 29851 df-wspthsn 29853 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |