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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 33782 | . . . . . . . 8 β’ ((πΊ β USGraph β§ π(CyclesβπΊ)π β§ π β β ) β 2 < (β―βπ)) |
3 | 2re 12232 | . . . . . . . . . . 11 β’ 2 β β | |
4 | 3 | rexri 11218 | . . . . . . . . . 10 β’ 2 β β* |
5 | 1 | fvexi 6857 | . . . . . . . . . . 11 β’ π β V |
6 | hashxrcl 14263 | . . . . . . . . . . 11 β’ (π β V β (β―βπ) β β*) | |
7 | 5, 6 | ax-mp 5 | . . . . . . . . . 10 β’ (β―βπ) β β* |
8 | xrltne 13088 | . . . . . . . . . 10 β’ ((2 β β* β§ (β―βπ) β β* β§ 2 < (β―βπ)) β (β―βπ) β 2) | |
9 | 4, 7, 8 | mp3an12 1452 | . . . . . . . . 9 β’ (2 < (β―βπ) β (β―βπ) β 2) |
10 | 9 | neneqd 2945 | . . . . . . . 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 1123 | . . . . . 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 408 | . . . 4 β’ ((πΊ β USGraph β§ (β―βπ) = 2) β Β¬ (π(CyclesβπΊ)π β§ π β β )) |
15 | 14 | nexdv 1940 | . . 3 β’ ((πΊ β USGraph β§ (β―βπ) = 2) β Β¬ βπ(π(CyclesβπΊ)π β§ π β β )) |
16 | 15 | nexdv 1940 | . 2 β’ ((πΊ β USGraph β§ (β―βπ) = 2) β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β )) |
17 | isacycgr 33796 | . . 3 β’ (πΊ β USGraph β (πΊ β AcyclicGraph β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ))) | |
18 | 17 | adantr 482 | . 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 397 β§ w3a 1088 = wceq 1542 βwex 1782 β wcel 2107 β wne 2940 Vcvv 3444 β c0 4283 class class class wbr 5106 βcfv 6497 β*cxr 11193 < clt 11194 2c2 12213 β―chash 14236 Vtxcvtx 27989 USGraphcusgr 28142 Cyclesccycls 28775 AcyclicGraphcacycgr 33793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9842 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-xnn0 12491 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-hash 14237 df-word 14409 df-edg 28041 df-uhgr 28051 df-upgr 28075 df-umgr 28076 df-uspgr 28143 df-usgr 28144 df-wlks 28589 df-trls 28682 df-pths 28706 df-crcts 28776 df-cycls 28777 df-acycgr 33794 |
This theorem is referenced by: (None) |
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