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Mirrors > Home > MPE Home > Th. List > Mathboxes > acycgr2v | Structured version Visualization version GIF version |
Description: A simple graph with two vertices is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.) |
Ref | Expression |
---|---|
acycgrv.1 | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
acycgr2v | ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → 𝐺 ∈ AcyclicGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acycgrv.1 | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | usgrcyclgt2v 32402 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → 2 < (♯‘𝑉)) |
3 | 2re 11709 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
4 | 3 | rexri 10696 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ* |
5 | 1 | fvexi 6681 | . . . . . . . . . . 11 ⊢ 𝑉 ∈ V |
6 | hashxrcl 13716 | . . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (♯‘𝑉) ∈ ℝ*) | |
7 | 5, 6 | ax-mp 5 | . . . . . . . . . 10 ⊢ (♯‘𝑉) ∈ ℝ* |
8 | xrltne 12554 | . . . . . . . . . 10 ⊢ ((2 ∈ ℝ* ∧ (♯‘𝑉) ∈ ℝ* ∧ 2 < (♯‘𝑉)) → (♯‘𝑉) ≠ 2) | |
9 | 4, 7, 8 | mp3an12 1446 | . . . . . . . . 9 ⊢ (2 < (♯‘𝑉) → (♯‘𝑉) ≠ 2) |
10 | 9 | neneqd 3020 | . . . . . . . 8 ⊢ (2 < (♯‘𝑉) → ¬ (♯‘𝑉) = 2) |
11 | 2, 10 | syl 17 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → ¬ (♯‘𝑉) = 2) |
12 | 11 | 3expib 1117 | . . . . . 6 ⊢ (𝐺 ∈ USGraph → ((𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅) → ¬ (♯‘𝑉) = 2)) |
13 | 12 | con2d 136 | . . . . 5 ⊢ (𝐺 ∈ USGraph → ((♯‘𝑉) = 2 → ¬ (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
14 | 13 | imp 409 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → ¬ (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
15 | 14 | nexdv 1936 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → ¬ ∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
16 | 15 | nexdv 1936 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)) |
17 | isacycgr 32416 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) | |
18 | 17 | adantr 483 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
19 | 16, 18 | mpbird 259 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → 𝐺 ∈ AcyclicGraph) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3015 Vcvv 3493 ∅c0 4288 class class class wbr 5063 ‘cfv 6352 ℝ*cxr 10671 < clt 10672 2c2 11690 ♯chash 13688 Vtxcvtx 26779 USGraphcusgr 26932 Cyclesccycls 27564 AcyclicGraphcacycgr 32413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-2o 8100 df-oadd 8103 df-er 8286 df-map 8405 df-pm 8406 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-dju 9327 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-n0 11896 df-xnn0 11966 df-z 11980 df-uz 12242 df-fz 12891 df-fzo 13032 df-hash 13689 df-word 13860 df-edg 26831 df-uhgr 26841 df-upgr 26865 df-umgr 26866 df-uspgr 26933 df-usgr 26934 df-wlks 27379 df-trls 27472 df-pths 27495 df-crcts 27565 df-cycls 27566 df-acycgr 32414 |
This theorem is referenced by: (None) |
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