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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 | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
acycgr2v | β’ ((πΊ β USGraph β§ (β―βπ) = 2) β πΊ β AcyclicGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acycgrv.1 | . . . . . . . . 9 β’ π = (VtxβπΊ) | |
2 | 1 | usgrcyclgt2v 34649 | . . . . . . . 8 β’ ((πΊ β USGraph β§ π(CyclesβπΊ)π β§ π β β ) β 2 < (β―βπ)) |
3 | 2re 12287 | . . . . . . . . . . 11 β’ 2 β β | |
4 | 3 | rexri 11273 | . . . . . . . . . 10 β’ 2 β β* |
5 | 1 | fvexi 6898 | . . . . . . . . . . 11 β’ π β V |
6 | hashxrcl 14319 | . . . . . . . . . . 11 β’ (π β V β (β―βπ) β β*) | |
7 | 5, 6 | ax-mp 5 | . . . . . . . . . 10 β’ (β―βπ) β β* |
8 | xrltne 13145 | . . . . . . . . . 10 β’ ((2 β β* β§ (β―βπ) β β* β§ 2 < (β―βπ)) β (β―βπ) β 2) | |
9 | 4, 7, 8 | mp3an12 1447 | . . . . . . . . 9 β’ (2 < (β―βπ) β (β―βπ) β 2) |
10 | 9 | neneqd 2939 | . . . . . . . 8 β’ (2 < (β―βπ) β Β¬ (β―βπ) = 2) |
11 | 2, 10 | syl 17 | . . . . . . 7 β’ ((πΊ β USGraph β§ π(CyclesβπΊ)π β§ π β β ) β Β¬ (β―βπ) = 2) |
12 | 11 | 3expib 1119 | . . . . . 6 β’ (πΊ β USGraph β ((π(CyclesβπΊ)π β§ π β β ) β Β¬ (β―βπ) = 2)) |
13 | 12 | con2d 134 | . . . . 5 β’ (πΊ β USGraph β ((β―βπ) = 2 β Β¬ (π(CyclesβπΊ)π β§ π β β ))) |
14 | 13 | imp 406 | . . . 4 β’ ((πΊ β USGraph β§ (β―βπ) = 2) β Β¬ (π(CyclesβπΊ)π β§ π β β )) |
15 | 14 | nexdv 1931 | . . 3 β’ ((πΊ β USGraph β§ (β―βπ) = 2) β Β¬ βπ(π(CyclesβπΊ)π β§ π β β )) |
16 | 15 | nexdv 1931 | . 2 β’ ((πΊ β USGraph β§ (β―βπ) = 2) β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β )) |
17 | isacycgr 34663 | . . 3 β’ (πΊ β USGraph β (πΊ β AcyclicGraph β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ))) | |
18 | 17 | adantr 480 | . 2 β’ ((πΊ β USGraph β§ (β―βπ) = 2) β (πΊ β AcyclicGraph β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ))) |
19 | 16, 18 | mpbird 257 | 1 β’ ((πΊ β USGraph β§ (β―βπ) = 2) β πΊ β AcyclicGraph) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 βwex 1773 β wcel 2098 β wne 2934 Vcvv 3468 β c0 4317 class class class wbr 5141 βcfv 6536 β*cxr 11248 < clt 11249 2c2 12268 β―chash 14292 Vtxcvtx 28759 USGraphcusgr 28912 Cyclesccycls 29546 AcyclicGraphcacycgr 34660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14293 df-word 14468 df-edg 28811 df-uhgr 28821 df-upgr 28845 df-umgr 28846 df-uspgr 28913 df-usgr 28914 df-wlks 29360 df-trls 29453 df-pths 29477 df-crcts 29547 df-cycls 29548 df-acycgr 34661 |
This theorem is referenced by: (None) |
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