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Mirrors > Home > MPE Home > Th. List > Mathboxes > cusgracyclt3v | Structured version Visualization version GIF version |
Description: A complete simple graph is acyclic if and only if it has fewer than three vertices. (Contributed by BTernaryTau, 20-Oct-2023.) |
Ref | Expression |
---|---|
cusgracyclt3v.1 | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
cusgracyclt3v | ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ (♯‘𝑉) < 3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isacycgr 32843 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
2 | 3nn0 12132 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
3 | cusgracyclt3v.1 | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | fvexi 6749 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
5 | hashxnn0 13929 | . . . . . . . 8 ⊢ (𝑉 ∈ V → (♯‘𝑉) ∈ ℕ0*) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℕ0* |
7 | xnn0lem1lt 12858 | . . . . . . 7 ⊢ ((3 ∈ ℕ0 ∧ (♯‘𝑉) ∈ ℕ0*) → (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉))) | |
8 | 2, 6, 7 | mp2an 692 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉)) |
9 | 3re 11934 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
10 | 9 | rexri 10915 | . . . . . . 7 ⊢ 3 ∈ ℝ* |
11 | xnn0xr 12191 | . . . . . . . 8 ⊢ ((♯‘𝑉) ∈ ℕ0* → (♯‘𝑉) ∈ ℝ*) | |
12 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℝ* |
13 | xrlenlt 10922 | . . . . . . 7 ⊢ ((3 ∈ ℝ* ∧ (♯‘𝑉) ∈ ℝ*) → (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3)) | |
14 | 10, 12, 13 | mp2an 692 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3) |
15 | 3m1e2 11982 | . . . . . . 7 ⊢ (3 − 1) = 2 | |
16 | 15 | breq1i 5074 | . . . . . 6 ⊢ ((3 − 1) < (♯‘𝑉) ↔ 2 < (♯‘𝑉)) |
17 | 8, 14, 16 | 3bitr3i 304 | . . . . 5 ⊢ (¬ (♯‘𝑉) < 3 ↔ 2 < (♯‘𝑉)) |
18 | 3 | cusgr3cyclex 32834 | . . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)) |
19 | 3ne0 11960 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
20 | neeq1 3004 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 3 → ((♯‘𝑓) ≠ 0 ↔ 3 ≠ 0)) | |
21 | 19, 20 | mpbiri 261 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) = 3 → (♯‘𝑓) ≠ 0) |
22 | hasheq0 13954 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ V → ((♯‘𝑓) = 0 ↔ 𝑓 = ∅)) | |
23 | 22 | elv 3426 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 0 ↔ 𝑓 = ∅) |
24 | 23 | necon3bii 2994 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) ≠ 0 ↔ 𝑓 ≠ ∅) |
25 | 21, 24 | sylib 221 | . . . . . . . . 9 ⊢ ((♯‘𝑓) = 3 → 𝑓 ≠ ∅) |
26 | 25 | anim2i 620 | . . . . . . . 8 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
27 | 26 | 2eximi 1843 | . . . . . . 7 ⊢ (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
28 | 18, 27 | syl 17 | . . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
29 | 28 | ex 416 | . . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → (2 < (♯‘𝑉) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
30 | 17, 29 | syl5bi 245 | . . . 4 ⊢ (𝐺 ∈ ComplUSGraph → (¬ (♯‘𝑉) < 3 → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
31 | 30 | con1d 147 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → (♯‘𝑉) < 3)) |
32 | 1, 31 | sylbid 243 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph → (♯‘𝑉) < 3)) |
33 | cusgrusgr 27531 | . . . . . . 7 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | |
34 | 3 | usgrcyclgt2v 32829 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → 2 < (♯‘𝑉)) |
35 | 34 | 3expib 1124 | . . . . . . 7 ⊢ (𝐺 ∈ USGraph → ((𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → 2 < (♯‘𝑉))) |
36 | 33, 35 | syl 17 | . . . . . 6 ⊢ (𝐺 ∈ ComplUSGraph → ((𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → 2 < (♯‘𝑉))) |
37 | 36, 17 | syl6ibr 255 | . . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → ((𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → ¬ (♯‘𝑉) < 3)) |
38 | 37 | exlimdvv 1942 | . . . 4 ⊢ (𝐺 ∈ ComplUSGraph → (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → ¬ (♯‘𝑉) < 3)) |
39 | 38 | con2d 136 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → ((♯‘𝑉) < 3 → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
40 | 39, 1 | sylibrd 262 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → ((♯‘𝑉) < 3 → 𝐺 ∈ AcyclicGraph)) |
41 | 32, 40 | impbid 215 | 1 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ (♯‘𝑉) < 3)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2111 ≠ wne 2941 Vcvv 3420 ∅c0 4251 class class class wbr 5067 ‘cfv 6397 (class class class)co 7231 0cc0 10753 1c1 10754 ℝ*cxr 10890 < clt 10891 ≤ cle 10892 − cmin 11086 2c2 11909 3c3 11910 ℕ0cn0 12114 ℕ0*cxnn0 12186 ♯chash 13920 Vtxcvtx 27111 USGraphcusgr 27264 ComplUSGraphccusgr 27522 Cyclesccycls 27896 AcyclicGraphcacycgr 32840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-2o 8223 df-oadd 8226 df-er 8411 df-map 8530 df-pm 8531 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-dju 9541 df-card 9579 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-nn 11855 df-2 11917 df-3 11918 df-4 11919 df-n0 12115 df-xnn0 12187 df-z 12201 df-uz 12463 df-xneg 12728 df-xadd 12729 df-fz 13120 df-fzo 13263 df-hash 13921 df-word 14094 df-concat 14150 df-s1 14177 df-s2 14437 df-s3 14438 df-s4 14439 df-edg 27163 df-uhgr 27173 df-upgr 27197 df-umgr 27198 df-uspgr 27265 df-usgr 27266 df-nbgr 27445 df-uvtx 27498 df-cplgr 27523 df-cusgr 27524 df-wlks 27711 df-trls 27804 df-pths 27827 df-crcts 27897 df-cycls 27898 df-acycgr 32841 |
This theorem is referenced by: (None) |
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