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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 32392 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
2 | 3nn0 11916 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
3 | cusgracyclt3v.1 | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | fvexi 6684 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
5 | hashxnn0 13700 | . . . . . . . 8 ⊢ (𝑉 ∈ V → (♯‘𝑉) ∈ ℕ0*) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℕ0* |
7 | xnn0lem1lt 12638 | . . . . . . 7 ⊢ ((3 ∈ ℕ0 ∧ (♯‘𝑉) ∈ ℕ0*) → (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉))) | |
8 | 2, 6, 7 | mp2an 690 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉)) |
9 | 3re 11718 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
10 | 9 | rexri 10699 | . . . . . . 7 ⊢ 3 ∈ ℝ* |
11 | xnn0xr 11973 | . . . . . . . 8 ⊢ ((♯‘𝑉) ∈ ℕ0* → (♯‘𝑉) ∈ ℝ*) | |
12 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℝ* |
13 | xrlenlt 10706 | . . . . . . 7 ⊢ ((3 ∈ ℝ* ∧ (♯‘𝑉) ∈ ℝ*) → (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3)) | |
14 | 10, 12, 13 | mp2an 690 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3) |
15 | 3m1e2 11766 | . . . . . . 7 ⊢ (3 − 1) = 2 | |
16 | 15 | breq1i 5073 | . . . . . 6 ⊢ ((3 − 1) < (♯‘𝑉) ↔ 2 < (♯‘𝑉)) |
17 | 8, 14, 16 | 3bitr3i 303 | . . . . 5 ⊢ (¬ (♯‘𝑉) < 3 ↔ 2 < (♯‘𝑉)) |
18 | 3 | cusgr3cyclex 32383 | . . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)) |
19 | 3ne0 11744 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
20 | neeq1 3078 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 3 → ((♯‘𝑓) ≠ 0 ↔ 3 ≠ 0)) | |
21 | 19, 20 | mpbiri 260 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) = 3 → (♯‘𝑓) ≠ 0) |
22 | hasheq0 13725 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ V → ((♯‘𝑓) = 0 ↔ 𝑓 = ∅)) | |
23 | 22 | elv 3499 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 0 ↔ 𝑓 = ∅) |
24 | 23 | necon3bii 3068 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) ≠ 0 ↔ 𝑓 ≠ ∅) |
25 | 21, 24 | sylib 220 | . . . . . . . . 9 ⊢ ((♯‘𝑓) = 3 → 𝑓 ≠ ∅) |
26 | 25 | anim2i 618 | . . . . . . . 8 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
27 | 26 | 2eximi 1836 | . . . . . . 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 415 | . . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → (2 < (♯‘𝑉) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
30 | 17, 29 | syl5bi 244 | . . . 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 242 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph → (♯‘𝑉) < 3)) |
33 | cusgrusgr 27201 | . . . . . . 7 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | |
34 | 3 | usgrcyclgt2v 32378 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → 2 < (♯‘𝑉)) |
35 | 34 | 3expib 1118 | . . . . . . 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 254 | . . . . 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 261 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → ((♯‘𝑉) < 3 → 𝐺 ∈ AcyclicGraph)) |
41 | 32, 40 | impbid 214 | 1 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ (♯‘𝑉) < 3)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 − cmin 10870 2c2 11693 3c3 11694 ℕ0cn0 11898 ℕ0*cxnn0 11968 ♯chash 13691 Vtxcvtx 26781 USGraphcusgr 26934 ComplUSGraphccusgr 27192 Cyclesccycls 27566 AcyclicGraphcacycgr 32389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-xneg 12508 df-xadd 12509 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-s4 14212 df-edg 26833 df-uhgr 26843 df-upgr 26867 df-umgr 26868 df-uspgr 26935 df-usgr 26936 df-nbgr 27115 df-uvtx 27168 df-cplgr 27193 df-cusgr 27194 df-wlks 27381 df-trls 27474 df-pths 27497 df-crcts 27567 df-cycls 27568 df-acycgr 32390 |
This theorem is referenced by: (None) |
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