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Mirrors > Home > MPE Home > Th. List > Mathboxes > usgrcyclgt2v | Structured version Visualization version GIF version |
Description: A simple graph with a non-trivial cycle must have at least 3 vertices. (Contributed by BTernaryTau, 5-Oct-2023.) |
Ref | Expression |
---|---|
usgrcyclgt2v.1 | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
usgrcyclgt2v | ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 11699 | . . . 4 ⊢ 2 ∈ ℝ | |
2 | 1 | rexri 10688 | . . 3 ⊢ 2 ∈ ℝ* |
3 | 2 | a1i 11 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 ∈ ℝ*) |
4 | cycliswlk 27587 | . . . . 5 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
5 | wlkcl 27405 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) |
7 | nn0xnn0 11959 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℕ0*) | |
8 | xnn0xr 11960 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0* → (♯‘𝐹) ∈ ℝ*) | |
9 | 6, 7, 8 | 3syl 18 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ∈ ℝ*) |
10 | 9 | 3ad2ant2 1131 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ∈ ℝ*) |
11 | usgrcyclgt2v.1 | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | 11 | fvexi 6659 | . . . 4 ⊢ 𝑉 ∈ V |
13 | hashxnn0 13695 | . . . 4 ⊢ (𝑉 ∈ V → (♯‘𝑉) ∈ ℕ0*) | |
14 | xnn0xr 11960 | . . . 4 ⊢ ((♯‘𝑉) ∈ ℕ0* → (♯‘𝑉) ∈ ℝ*) | |
15 | 12, 13, 14 | mp2b 10 | . . 3 ⊢ (♯‘𝑉) ∈ ℝ* |
16 | 15 | a1i 11 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (♯‘𝑉) ∈ ℝ*) |
17 | usgrgt2cycl 32490 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝐹)) | |
18 | cyclispth 27586 | . . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃) | |
19 | 11 | pthhashvtx 32487 | . . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉)) |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉)) |
21 | 20 | 3ad2ant2 1131 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≤ (♯‘𝑉)) |
22 | 3, 10, 16, 17, 21 | xrltletrd 12542 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 2c2 11680 ℕ0cn0 11885 ℕ0*cxnn0 11955 ♯chash 13686 Vtxcvtx 26789 USGraphcusgr 26942 Walkscwlks 27386 Pathscpths 27501 Cyclesccycls 27574 |
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-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-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-edg 26841 df-uhgr 26851 df-upgr 26875 df-umgr 26876 df-uspgr 26943 df-usgr 26944 df-wlks 27389 df-trls 27482 df-pths 27505 df-crcts 27575 df-cycls 27576 |
This theorem is referenced by: acycgr2v 32510 cusgracyclt3v 32516 |
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