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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 35130 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
2 | 3nn0 12542 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
3 | cusgracyclt3v.1 | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | fvexi 6921 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
5 | hashxnn0 14375 | . . . . . . . 8 ⊢ (𝑉 ∈ V → (♯‘𝑉) ∈ ℕ0*) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℕ0* |
7 | xnn0lem1lt 13283 | . . . . . . 7 ⊢ ((3 ∈ ℕ0 ∧ (♯‘𝑉) ∈ ℕ0*) → (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉))) | |
8 | 2, 6, 7 | mp2an 692 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ (3 − 1) < (♯‘𝑉)) |
9 | 3re 12344 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
10 | 9 | rexri 11317 | . . . . . . 7 ⊢ 3 ∈ ℝ* |
11 | xnn0xr 12602 | . . . . . . . 8 ⊢ ((♯‘𝑉) ∈ ℕ0* → (♯‘𝑉) ∈ ℝ*) | |
12 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝑉) ∈ ℝ* |
13 | xrlenlt 11324 | . . . . . . 7 ⊢ ((3 ∈ ℝ* ∧ (♯‘𝑉) ∈ ℝ*) → (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3)) | |
14 | 10, 12, 13 | mp2an 692 | . . . . . 6 ⊢ (3 ≤ (♯‘𝑉) ↔ ¬ (♯‘𝑉) < 3) |
15 | 3m1e2 12392 | . . . . . . 7 ⊢ (3 − 1) = 2 | |
16 | 15 | breq1i 5155 | . . . . . 6 ⊢ ((3 − 1) < (♯‘𝑉) ↔ 2 < (♯‘𝑉)) |
17 | 8, 14, 16 | 3bitr3i 301 | . . . . 5 ⊢ (¬ (♯‘𝑉) < 3 ↔ 2 < (♯‘𝑉)) |
18 | 3 | cusgr3cyclex 35121 | . . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)) |
19 | 3ne0 12370 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
20 | neeq1 3001 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 3 → ((♯‘𝑓) ≠ 0 ↔ 3 ≠ 0)) | |
21 | 19, 20 | mpbiri 258 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) = 3 → (♯‘𝑓) ≠ 0) |
22 | hasheq0 14399 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ V → ((♯‘𝑓) = 0 ↔ 𝑓 = ∅)) | |
23 | 22 | elv 3483 | . . . . . . . . . . 11 ⊢ ((♯‘𝑓) = 0 ↔ 𝑓 = ∅) |
24 | 23 | necon3bii 2991 | . . . . . . . . . 10 ⊢ ((♯‘𝑓) ≠ 0 ↔ 𝑓 ≠ ∅) |
25 | 21, 24 | sylib 218 | . . . . . . . . 9 ⊢ ((♯‘𝑓) = 3 → 𝑓 ≠ ∅) |
26 | 25 | anim2i 617 | . . . . . . . 8 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
27 | 26 | 2eximi 1833 | . . . . . . 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 412 | . . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → (2 < (♯‘𝑉) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
30 | 17, 29 | biimtrid 242 | . . . 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 240 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph → (♯‘𝑉) < 3)) |
33 | cusgrusgr 29451 | . . . . . . 7 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph) | |
34 | 3 | usgrcyclgt2v 35116 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → 2 < (♯‘𝑉)) |
35 | 34 | 3expib 1121 | . . . . . . 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 252 | . . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → ((𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → ¬ (♯‘𝑉) < 3)) |
38 | 37 | exlimdvv 1932 | . . . 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 259 | . 2 ⊢ (𝐺 ∈ ComplUSGraph → ((♯‘𝑉) < 3 → 𝐺 ∈ AcyclicGraph)) |
41 | 32, 40 | impbid 212 | 1 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ (♯‘𝑉) < 3)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 2c2 12319 3c3 12320 ℕ0cn0 12524 ℕ0*cxnn0 12597 ♯chash 14366 Vtxcvtx 29028 USGraphcusgr 29181 ComplUSGraphccusgr 29442 Cyclesccycls 29818 AcyclicGraphcacycgr 35127 |
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-ifp 1063 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-tp 4636 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-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 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-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-xneg 13152 df-xadd 13153 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-s3 14885 df-s4 14886 df-edg 29080 df-uhgr 29090 df-upgr 29114 df-umgr 29115 df-uspgr 29182 df-usgr 29183 df-nbgr 29365 df-uvtx 29418 df-cplgr 29443 df-cusgr 29444 df-wlks 29632 df-trls 29725 df-pths 29749 df-crcts 29819 df-cycls 29820 df-acycgr 35128 |
This theorem is referenced by: (None) |
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