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Mirrors > Home > MPE Home > Th. List > clwwlknon1le1 | Structured version Visualization version GIF version |
Description: There is at most one (closed) walk on vertex 𝑋 of length 1 as word over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknon1le1 | ⊢ (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2799 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2799 | . . . . 5 ⊢ (ClWWalksNOn‘𝐺) = (ClWWalksNOn‘𝐺) | |
3 | eqid 2799 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 1, 2, 3 | clwwlknon1loop 27437 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∈ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉}) |
5 | fveq2 6411 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘{〈“𝑋”〉})) | |
6 | s1cli 13625 | . . . . . . 7 ⊢ 〈“𝑋”〉 ∈ Word V | |
7 | hashsng 13409 | . . . . . . 7 ⊢ (〈“𝑋”〉 ∈ Word V → (♯‘{〈“𝑋”〉}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{〈“𝑋”〉}) = 1 |
9 | 5, 8 | syl6eq 2849 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 1) |
10 | 1le1 10947 | . . . . 5 ⊢ 1 ≤ 1 | |
11 | 9, 10 | syl6eqbr 4882 | . . . 4 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
12 | 4, 11 | syl 17 | . . 3 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∈ (Edg‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
13 | 1, 2, 3 | clwwlknon1nloop 27438 | . . . . 5 ⊢ ({𝑋} ∉ (Edg‘𝐺) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
14 | 13 | adantl 474 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∉ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
15 | fveq2 6411 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) | |
16 | hash0 13408 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
17 | 15, 16 | syl6eq 2849 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
18 | 0le1 10843 | . . . . 5 ⊢ 0 ≤ 1 | |
19 | 17, 18 | syl6eqbr 4882 | . . . 4 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
20 | 14, 19 | syl 17 | . . 3 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∉ (Edg‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
21 | 12, 20 | pm2.61danel 3088 | . 2 ⊢ (𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
22 | id 22 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (Vtx‘𝐺)) | |
23 | 22 | intnanrd 484 | . . . . . 6 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ)) |
24 | clwwlknon0 27431 | . . . . . 6 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) | |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
26 | 25 | fveq2d 6415 | . . . 4 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) |
27 | 26, 16 | syl6eq 2849 | . . 3 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
28 | 27, 18 | syl6eqbr 4882 | . 2 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
29 | 21, 28 | pm2.61i 177 | 1 ⊢ (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∉ wnel 3074 Vcvv 3385 ∅c0 4115 {csn 4368 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 0cc0 10224 1c1 10225 ≤ cle 10364 ℕcn 11312 ♯chash 13370 Word cword 13534 〈“cs1 13615 Vtxcvtx 26231 Edgcedg 26282 ClWWalksNOncclwwlknon 27423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-xnn0 11653 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535 df-lsw 13583 df-s1 13616 df-clwwlk 27275 df-clwwlkn 27328 df-clwwlknon 27424 |
This theorem is referenced by: (None) |
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