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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 2725 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2725 | . . . . 5 ⊢ (ClWWalksNOn‘𝐺) = (ClWWalksNOn‘𝐺) | |
3 | eqid 2725 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 1, 2, 3 | clwwlknon1loop 29985 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∈ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉}) |
5 | fveq2 6896 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘{〈“𝑋”〉})) | |
6 | s1cli 14596 | . . . . . . 7 ⊢ 〈“𝑋”〉 ∈ Word V | |
7 | hashsng 14369 | . . . . . . 7 ⊢ (〈“𝑋”〉 ∈ Word V → (♯‘{〈“𝑋”〉}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{〈“𝑋”〉}) = 1 |
9 | 5, 8 | eqtrdi 2781 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 1) |
10 | 1le1 11879 | . . . . 5 ⊢ 1 ≤ 1 | |
11 | 9, 10 | eqbrtrdi 5188 | . . . 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 29986 | . . . . 5 ⊢ ({𝑋} ∉ (Edg‘𝐺) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
14 | 13 | adantl 480 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∉ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
15 | fveq2 6896 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) | |
16 | hash0 14367 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
17 | 15, 16 | eqtrdi 2781 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
18 | 0le1 11774 | . . . . 5 ⊢ 0 ≤ 1 | |
19 | 17, 18 | eqbrtrdi 5188 | . . . 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 3049 | . 2 ⊢ (𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
22 | id 22 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (Vtx‘𝐺)) | |
23 | 22 | intnanrd 488 | . . . . . 6 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ)) |
24 | clwwlknon0 29980 | . . . . . 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 6900 | . . . 4 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) |
27 | 26, 16 | eqtrdi 2781 | . . 3 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
28 | 27, 18 | eqbrtrdi 5188 | . 2 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
29 | 21, 28 | pm2.61i 182 | 1 ⊢ (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∉ wnel 3035 Vcvv 3461 ∅c0 4322 {csn 4630 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 0cc0 11145 1c1 11146 ≤ cle 11286 ℕcn 12250 ♯chash 14330 Word cword 14505 〈“cs1 14586 Vtxcvtx 28886 Edgcedg 28937 ClWWalksNOncclwwlknon 29974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14331 df-word 14506 df-lsw 14554 df-s1 14587 df-clwwlk 29869 df-clwwlkn 29912 df-clwwlknon 29975 |
This theorem is referenced by: (None) |
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