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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 2739 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2739 | . . . . 5 ⊢ (ClWWalksNOn‘𝐺) = (ClWWalksNOn‘𝐺) | |
3 | eqid 2739 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 1, 2, 3 | clwwlknon1loop 28048 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∈ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉}) |
5 | fveq2 6687 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘{〈“𝑋”〉})) | |
6 | s1cli 14061 | . . . . . . 7 ⊢ 〈“𝑋”〉 ∈ Word V | |
7 | hashsng 13835 | . . . . . . 7 ⊢ (〈“𝑋”〉 ∈ Word V → (♯‘{〈“𝑋”〉}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{〈“𝑋”〉}) = 1 |
9 | 5, 8 | eqtrdi 2790 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 1) |
10 | 1le1 11359 | . . . . 5 ⊢ 1 ≤ 1 | |
11 | 9, 10 | eqbrtrdi 5079 | . . . 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 28049 | . . . . 5 ⊢ ({𝑋} ∉ (Edg‘𝐺) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
14 | 13 | adantl 485 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∉ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
15 | fveq2 6687 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) | |
16 | hash0 13833 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
17 | 15, 16 | eqtrdi 2790 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
18 | 0le1 11254 | . . . . 5 ⊢ 0 ≤ 1 | |
19 | 17, 18 | eqbrtrdi 5079 | . . . 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 3053 | . 2 ⊢ (𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
22 | id 22 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (Vtx‘𝐺)) | |
23 | 22 | intnanrd 493 | . . . . . 6 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ)) |
24 | clwwlknon0 28043 | . . . . . 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 6691 | . . . 4 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) |
27 | 26, 16 | eqtrdi 2790 | . . 3 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
28 | 27, 18 | eqbrtrdi 5079 | . 2 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
29 | 21, 28 | pm2.61i 185 | 1 ⊢ (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∉ wnel 3039 Vcvv 3400 ∅c0 4221 {csn 4526 class class class wbr 5040 ‘cfv 6350 (class class class)co 7183 0cc0 10628 1c1 10629 ≤ cle 10767 ℕcn 11729 ♯chash 13795 Word cword 13968 〈“cs1 14051 Vtxcvtx 26954 Edgcedg 27005 ClWWalksNOncclwwlknon 28037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-oadd 8148 df-er 8333 df-map 8452 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-n0 11990 df-xnn0 12062 df-z 12076 df-uz 12338 df-fz 12995 df-fzo 13138 df-hash 13796 df-word 13969 df-lsw 14017 df-s1 14052 df-clwwlk 27932 df-clwwlkn 27975 df-clwwlknon 28038 |
This theorem is referenced by: (None) |
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