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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 32505 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
2 | 3nn0 11903 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
3 | cusgracyclt3v.1 | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | fvexi 6659 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
5 | hashxnn0 13695 | . . . . . . . 8 ⊢ (𝑉 ∈ V → (♯‘𝑉) ∈ ℕ0*) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℕ0* |
7 | xnn0lem1lt 12625 | . . . . . . 7 ⊢ ((3 ∈ ℕ0 ∧ (♯‘𝑉) ∈ ℕ0*) → (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉))) | |
8 | 2, 6, 7 | mp2an 691 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉)) |
9 | 3re 11705 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
10 | 9 | rexri 10688 | . . . . . . 7 ⊢ 3 ∈ ℝ* |
11 | xnn0xr 11960 | . . . . . . . 8 ⊢ ((♯‘𝑉) ∈ ℕ0* → (♯‘𝑉) ∈ ℝ*) | |
12 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℝ* |
13 | xrlenlt 10695 | . . . . . . 7 ⊢ ((3 ∈ ℝ* ∧ (♯‘𝑉) ∈ ℝ*) → (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3)) | |
14 | 10, 12, 13 | mp2an 691 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3) |
15 | 3m1e2 11753 | . . . . . . 7 ⊢ (3 − 1) = 2 | |
16 | 15 | breq1i 5037 | . . . . . 6 ⊢ ((3 − 1) < (♯‘𝑉) ↔ 2 < (♯‘𝑉)) |
17 | 8, 14, 16 | 3bitr3i 304 | . . . . 5 ⊢ (¬ (♯‘𝑉) < 3 ↔ 2 < (♯‘𝑉)) |
18 | 3 | cusgr3cyclex 32496 | . . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)) |
19 | 3ne0 11731 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
20 | neeq1 3049 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 3 → ((♯‘𝑓) ≠ 0 ↔ 3 ≠ 0)) | |
21 | 19, 20 | mpbiri 261 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) = 3 → (♯‘𝑓) ≠ 0) |
22 | hasheq0 13720 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ V → ((♯‘𝑓) = 0 ↔ 𝑓 = ∅)) | |
23 | 22 | elv 3446 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 0 ↔ 𝑓 = ∅) |
24 | 23 | necon3bii 3039 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) ≠ 0 ↔ 𝑓 ≠ ∅) |
25 | 21, 24 | sylib 221 | . . . . . . . . 9 ⊢ ((♯‘𝑓) = 3 → 𝑓 ≠ ∅) |
26 | 25 | anim2i 619 | . . . . . . . 8 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
27 | 26 | 2eximi 1837 | . . . . . . 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 27209 | . . . . . . 7 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | |
34 | 3 | usgrcyclgt2v 32491 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → 2 < (♯‘𝑉)) |
35 | 34 | 3expib 1119 | . . . . . . 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 1935 | . . . 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 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 2c2 11680 3c3 11681 ℕ0cn0 11885 ℕ0*cxnn0 11955 ♯chash 13686 Vtxcvtx 26789 USGraphcusgr 26942 ComplUSGraphccusgr 27200 Cyclesccycls 27574 AcyclicGraphcacycgr 32502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xneg 12495 df-xadd 12496 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-s4 14203 df-edg 26841 df-uhgr 26851 df-upgr 26875 df-umgr 26876 df-uspgr 26943 df-usgr 26944 df-nbgr 27123 df-uvtx 27176 df-cplgr 27201 df-cusgr 27202 df-wlks 27389 df-trls 27482 df-pths 27505 df-crcts 27575 df-cycls 27576 df-acycgr 32503 |
This theorem is referenced by: (None) |
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