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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 11705 | . . . 4 ⊢ 2 ∈ ℝ | |
2 | 1 | rexri 10692 | . . 3 ⊢ 2 ∈ ℝ* |
3 | 2 | a1i 11 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 ∈ ℝ*) |
4 | cycliswlk 27575 | . . . . 5 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
5 | wlkcl 27393 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) |
7 | nn0xnn0 11965 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℕ0*) | |
8 | xnn0xr 11966 | . . . 4 ⊢ ((♯‘𝐹) ∈ ℕ0* → (♯‘𝐹) ∈ ℝ*) | |
9 | 6, 7, 8 | 3syl 18 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ∈ ℝ*) |
10 | 9 | 3ad2ant2 1129 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ∈ ℝ*) |
11 | usgrcyclgt2v.1 | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | 11 | fvexi 6677 | . . . 4 ⊢ 𝑉 ∈ V |
13 | hashxnn0 13696 | . . . 4 ⊢ (𝑉 ∈ V → (♯‘𝑉) ∈ ℕ0*) | |
14 | xnn0xr 11966 | . . . 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 32396 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝐹)) | |
18 | cyclispth 27574 | . . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃) | |
19 | 11 | pthhashvtx 32393 | . . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉)) |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉)) |
21 | 20 | 3ad2ant2 1129 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → (♯‘𝐹) ≤ (♯‘𝑉)) |
22 | 3, 10, 16, 17, 21 | xrltletrd 12548 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 Vcvv 3491 ∅c0 4284 class class class wbr 5059 ‘cfv 6348 ℝ*cxr 10667 < clt 10668 ≤ cle 10669 2c2 11686 ℕ0cn0 11891 ℕ0*cxnn0 11961 ♯chash 13687 Vtxcvtx 26777 USGraphcusgr 26930 Walkscwlks 27374 Pathscpths 27489 Cyclesccycls 27562 |
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 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
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 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-dju 9323 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-edg 26829 df-uhgr 26839 df-upgr 26863 df-umgr 26864 df-uspgr 26931 df-usgr 26932 df-wlks 27377 df-trls 27470 df-pths 27493 df-crcts 27563 df-cycls 27564 |
This theorem is referenced by: acycgr2v 32416 cusgracyclt3v 32422 |
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