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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 11712 | . . . 4 ⊢ 2 ∈ ℝ | |
2 | 1 | rexri 10699 | . . 3 ⊢ 2 ∈ ℝ* |
3 | 2 | a1i 11 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 ∈ ℝ*) |
4 | cycliswlk 27579 | . . . . 5 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
5 | wlkcl 27397 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) |
7 | nn0xnn0 11972 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℕ0*) | |
8 | xnn0xr 11973 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0* → (♯‘𝐹) ∈ ℝ*) | |
9 | 6, 7, 8 | 3syl 18 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ∈ ℝ*) |
10 | 9 | 3ad2ant2 1130 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ∈ ℝ*) |
11 | usgrcyclgt2v.1 | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | 11 | fvexi 6684 | . . . 4 ⊢ 𝑉 ∈ V |
13 | hashxnn0 13700 | . . . 4 ⊢ (𝑉 ∈ V → (♯‘𝑉) ∈ ℕ0*) | |
14 | xnn0xr 11973 | . . . 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 32377 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝐹)) | |
18 | cyclispth 27578 | . . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃) | |
19 | 11 | pthhashvtx 32374 | . . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉)) |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉)) |
21 | 20 | 3ad2ant2 1130 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≤ (♯‘𝑉)) |
22 | 3, 10, 16, 17, 21 | xrltletrd 12555 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 2c2 11693 ℕ0cn0 11898 ℕ0*cxnn0 11968 ♯chash 13691 Vtxcvtx 26781 USGraphcusgr 26934 Walkscwlks 27378 Pathscpths 27493 Cyclesccycls 27566 |
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-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-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-edg 26833 df-uhgr 26843 df-upgr 26867 df-umgr 26868 df-uspgr 26935 df-usgr 26936 df-wlks 27381 df-trls 27474 df-pths 27497 df-crcts 27567 df-cycls 27568 |
This theorem is referenced by: acycgr2v 32397 cusgracyclt3v 32403 |
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