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Mirrors > Home > MPE Home > Th. List > clwwlknon0 | Structured version Visualization version GIF version |
Description: Sufficient conditions for ClWWalksNOn to be empty. (Contributed by AV, 25-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknon0 | ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6930 | . . . 4 ⊢ (𝑁 = 0 → (𝑋(ClWWalksNOn‘𝐺)𝑁) = (𝑋(ClWWalksNOn‘𝐺)0)) | |
2 | clwwlk0on0 27494 | . . . 4 ⊢ (𝑋(ClWWalksNOn‘𝐺)0) = ∅ | |
3 | 1, 2 | syl6eq 2829 | . . 3 ⊢ (𝑁 = 0 → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅) |
4 | 3 | a1d 25 | . 2 ⊢ (𝑁 = 0 → (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅)) |
5 | simprl 761 | . . . . . 6 ⊢ ((𝑁 ≠ 0 ∧ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0)) → 𝑋 ∈ (Vtx‘𝐺)) | |
6 | elnnne0 11658 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) | |
7 | 6 | simplbi2 496 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≠ 0 → 𝑁 ∈ ℕ)) |
8 | 7 | adantl 475 | . . . . . . 7 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑁 ≠ 0 → 𝑁 ∈ ℕ)) |
9 | 8 | impcom 398 | . . . . . 6 ⊢ ((𝑁 ≠ 0 ∧ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℕ) |
10 | 5, 9 | jca 507 | . . . . 5 ⊢ ((𝑁 ≠ 0 ∧ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0)) → (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ)) |
11 | 10 | stoic1a 1816 | . . . 4 ⊢ ((𝑁 ≠ 0 ∧ ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ)) → ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0)) |
12 | clwwlknonmpt2 27491 | . . . . 5 ⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) | |
13 | 12 | mpt2ndm0 7152 | . . . 4 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅) |
14 | 11, 13 | syl 17 | . . 3 ⊢ ((𝑁 ≠ 0 ∧ ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ)) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅) |
15 | 14 | ex 403 | . 2 ⊢ (𝑁 ≠ 0 → (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅)) |
16 | 4, 15 | pm2.61ine 3052 | 1 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 {crab 3093 ∅c0 4140 ‘cfv 6135 (class class class)co 6922 0cc0 10272 ℕcn 11374 ℕ0cn0 11642 Vtxcvtx 26344 ClWWalksN cclwwlkn 27413 ClWWalksNOncclwwlknon 27489 |
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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-clwwlk 27362 df-clwwlkn 27414 df-clwwlknon 27490 |
This theorem is referenced by: clwwlknon1nloop 27501 clwwlknon1le1 27503 |
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