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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 34888 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
2 | 3nn0 12528 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
3 | cusgracyclt3v.1 | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | fvexi 6910 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
5 | hashxnn0 14339 | . . . . . . . 8 ⊢ (𝑉 ∈ V → (♯‘𝑉) ∈ ℕ0*) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℕ0* |
7 | xnn0lem1lt 13263 | . . . . . . 7 ⊢ ((3 ∈ ℕ0 ∧ (♯‘𝑉) ∈ ℕ0*) → (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉))) | |
8 | 2, 6, 7 | mp2an 690 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉)) |
9 | 3re 12330 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
10 | 9 | rexri 11309 | . . . . . . 7 ⊢ 3 ∈ ℝ* |
11 | xnn0xr 12587 | . . . . . . . 8 ⊢ ((♯‘𝑉) ∈ ℕ0* → (♯‘𝑉) ∈ ℝ*) | |
12 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℝ* |
13 | xrlenlt 11316 | . . . . . . 7 ⊢ ((3 ∈ ℝ* ∧ (♯‘𝑉) ∈ ℝ*) → (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3)) | |
14 | 10, 12, 13 | mp2an 690 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3) |
15 | 3m1e2 12378 | . . . . . . 7 ⊢ (3 − 1) = 2 | |
16 | 15 | breq1i 5156 | . . . . . 6 ⊢ ((3 − 1) < (♯‘𝑉) ↔ 2 < (♯‘𝑉)) |
17 | 8, 14, 16 | 3bitr3i 300 | . . . . 5 ⊢ (¬ (♯‘𝑉) < 3 ↔ 2 < (♯‘𝑉)) |
18 | 3 | cusgr3cyclex 34879 | . . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)) |
19 | 3ne0 12356 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
20 | neeq1 2992 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 3 → ((♯‘𝑓) ≠ 0 ↔ 3 ≠ 0)) | |
21 | 19, 20 | mpbiri 257 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) = 3 → (♯‘𝑓) ≠ 0) |
22 | hasheq0 14363 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ V → ((♯‘𝑓) = 0 ↔ 𝑓 = ∅)) | |
23 | 22 | elv 3467 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 0 ↔ 𝑓 = ∅) |
24 | 23 | necon3bii 2982 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) ≠ 0 ↔ 𝑓 ≠ ∅) |
25 | 21, 24 | sylib 217 | . . . . . . . . 9 ⊢ ((♯‘𝑓) = 3 → 𝑓 ≠ ∅) |
26 | 25 | anim2i 615 | . . . . . . . 8 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
27 | 26 | 2eximi 1830 | . . . . . . 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 411 | . . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → (2 < (♯‘𝑉) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
30 | 17, 29 | biimtrid 241 | . . . 4 ⊢ (𝐺 ∈ ComplUSGraph → (¬ (♯‘𝑉) < 3 → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
31 | 30 | con1d 145 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → (♯‘𝑉) < 3)) |
32 | 1, 31 | sylbid 239 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph → (♯‘𝑉) < 3)) |
33 | cusgrusgr 29309 | . . . . . . 7 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | |
34 | 3 | usgrcyclgt2v 34874 | . . . . . . . 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 | imbitrrdi 251 | . . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → ((𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → ¬ (♯‘𝑉) < 3)) |
38 | 37 | exlimdvv 1929 | . . . 4 ⊢ (𝐺 ∈ ComplUSGraph → (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → ¬ (♯‘𝑉) < 3)) |
39 | 38 | con2d 134 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → ((♯‘𝑉) < 3 → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
40 | 39, 1 | sylibrd 258 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → ((♯‘𝑉) < 3 → 𝐺 ∈ AcyclicGraph)) |
41 | 32, 40 | impbid 211 | 1 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ (♯‘𝑉) < 3)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ∅c0 4322 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 0cc0 11145 1c1 11146 ℝ*cxr 11284 < clt 11285 ≤ cle 11286 − cmin 11481 2c2 12305 3c3 12306 ℕ0cn0 12510 ℕ0*cxnn0 12582 ♯chash 14330 Vtxcvtx 28886 USGraphcusgr 29039 ComplUSGraphccusgr 29300 Cyclesccycls 29676 AcyclicGraphcacycgr 34885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9931 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-xneg 13132 df-xadd 13133 df-fz 13525 df-fzo 13668 df-hash 14331 df-word 14506 df-concat 14562 df-s1 14587 df-s2 14840 df-s3 14841 df-s4 14842 df-edg 28938 df-uhgr 28948 df-upgr 28972 df-umgr 28973 df-uspgr 29040 df-usgr 29041 df-nbgr 29223 df-uvtx 29276 df-cplgr 29301 df-cusgr 29302 df-wlks 29490 df-trls 29583 df-pths 29607 df-crcts 29677 df-cycls 29678 df-acycgr 34886 |
This theorem is referenced by: (None) |
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