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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 2735 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2735 | . . . . 5 ⊢ (ClWWalksNOn‘𝐺) = (ClWWalksNOn‘𝐺) | |
3 | eqid 2735 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 1, 2, 3 | clwwlknon1loop 30127 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∈ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉}) |
5 | fveq2 6907 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘{〈“𝑋”〉})) | |
6 | s1cli 14640 | . . . . . . 7 ⊢ 〈“𝑋”〉 ∈ Word V | |
7 | hashsng 14405 | . . . . . . 7 ⊢ (〈“𝑋”〉 ∈ Word V → (♯‘{〈“𝑋”〉}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (♯‘{〈“𝑋”〉}) = 1 |
9 | 5, 8 | eqtrdi 2791 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = {〈“𝑋”〉} → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 1) |
10 | 1le1 11889 | . . . . 5 ⊢ 1 ≤ 1 | |
11 | 9, 10 | eqbrtrdi 5187 | . . . 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 30128 | . . . . 5 ⊢ ({𝑋} ∉ (Edg‘𝐺) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
14 | 13 | adantl 481 | . . . 4 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ {𝑋} ∉ (Edg‘𝐺)) → (𝑋(ClWWalksNOn‘𝐺)1) = ∅) |
15 | fveq2 6907 | . . . . . 6 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) | |
16 | hash0 14403 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
17 | 15, 16 | eqtrdi 2791 | . . . . 5 ⊢ ((𝑋(ClWWalksNOn‘𝐺)1) = ∅ → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
18 | 0le1 11784 | . . . . 5 ⊢ 0 ≤ 1 | |
19 | 17, 18 | eqbrtrdi 5187 | . . . 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 3058 | . 2 ⊢ (𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1) |
22 | id 22 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (Vtx‘𝐺)) | |
23 | 22 | intnanrd 489 | . . . . . 6 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 1 ∈ ℕ)) |
24 | clwwlknon0 30122 | . . . . . 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 6911 | . . . 4 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = (♯‘∅)) |
27 | 26, 16 | eqtrdi 2791 | . . 3 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) = 0) |
28 | 27, 18 | eqbrtrdi 5187 | . 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 395 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 Vcvv 3478 ∅c0 4339 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 ≤ cle 11294 ℕcn 12264 ♯chash 14366 Word cword 14549 〈“cs1 14630 Vtxcvtx 29028 Edgcedg 29079 ClWWalksNOncclwwlknon 30116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-lsw 14598 df-s1 14631 df-clwwlk 30011 df-clwwlkn 30054 df-clwwlknon 30117 |
This theorem is referenced by: (None) |
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