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Mirrors > Home > MPE Home > Th. List > Mathboxes > acycgrislfgr | Structured version Visualization version GIF version |
Description: An acyclic hypergraph is a loop-free hypergraph. (Contributed by BTernaryTau, 15-Oct-2023.) |
Ref | Expression |
---|---|
acycgrislfgr.1 | β’ π = (VtxβπΊ) |
acycgrislfgr.2 | β’ πΌ = (iEdgβπΊ) |
Ref | Expression |
---|---|
acycgrislfgr | β’ ((πΊ β AcyclicGraph β§ πΊ β UHGraph) β πΌ:dom πΌβΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isacycgr 34205 | . . . 4 β’ (πΊ β UHGraph β (πΊ β AcyclicGraph β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ))) | |
2 | 1 | biimpac 479 | . . 3 β’ ((πΊ β AcyclicGraph β§ πΊ β UHGraph) β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β )) |
3 | loop1cycl 34197 | . . . . . . . 8 β’ (πΊ β UHGraph β (βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 1 β§ (πβ0) = π) β {π} β (EdgβπΊ))) | |
4 | 3simpa 1148 | . . . . . . . . 9 β’ ((π(CyclesβπΊ)π β§ (β―βπ) = 1 β§ (πβ0) = π) β (π(CyclesβπΊ)π β§ (β―βπ) = 1)) | |
5 | 4 | 2eximi 1838 | . . . . . . . 8 β’ (βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 1 β§ (πβ0) = π) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 1)) |
6 | 3, 5 | syl6bir 253 | . . . . . . 7 β’ (πΊ β UHGraph β ({π} β (EdgβπΊ) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 1))) |
7 | 6 | exlimdv 1936 | . . . . . 6 β’ (πΊ β UHGraph β (βπ{π} β (EdgβπΊ) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 1))) |
8 | vex 3478 | . . . . . . . . 9 β’ π β V | |
9 | hash1n0 14383 | . . . . . . . . 9 β’ ((π β V β§ (β―βπ) = 1) β π β β ) | |
10 | 8, 9 | mpan 688 | . . . . . . . 8 β’ ((β―βπ) = 1 β π β β ) |
11 | 10 | anim2i 617 | . . . . . . 7 β’ ((π(CyclesβπΊ)π β§ (β―βπ) = 1) β (π(CyclesβπΊ)π β§ π β β )) |
12 | 11 | 2eximi 1838 | . . . . . 6 β’ (βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 1) β βπβπ(π(CyclesβπΊ)π β§ π β β )) |
13 | 7, 12 | syl6 35 | . . . . 5 β’ (πΊ β UHGraph β (βπ{π} β (EdgβπΊ) β βπβπ(π(CyclesβπΊ)π β§ π β β ))) |
14 | 13 | con3d 152 | . . . 4 β’ (πΊ β UHGraph β (Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ) β Β¬ βπ{π} β (EdgβπΊ))) |
15 | 14 | adantl 482 | . . 3 β’ ((πΊ β AcyclicGraph β§ πΊ β UHGraph) β (Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ) β Β¬ βπ{π} β (EdgβπΊ))) |
16 | 2, 15 | mpd 15 | . 2 β’ ((πΊ β AcyclicGraph β§ πΊ β UHGraph) β Β¬ βπ{π} β (EdgβπΊ)) |
17 | acycgrislfgr.1 | . . . 4 β’ π = (VtxβπΊ) | |
18 | acycgrislfgr.2 | . . . 4 β’ πΌ = (iEdgβπΊ) | |
19 | 17, 18 | lfuhgr3 34179 | . . 3 β’ (πΊ β UHGraph β (πΌ:dom πΌβΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β Β¬ βπ{π} β (EdgβπΊ))) |
20 | 19 | adantl 482 | . 2 β’ ((πΊ β AcyclicGraph β§ πΊ β UHGraph) β (πΌ:dom πΌβΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)} β Β¬ βπ{π} β (EdgβπΊ))) |
21 | 16, 20 | mpbird 256 | 1 β’ ((πΊ β AcyclicGraph β§ πΊ β UHGraph) β πΌ:dom πΌβΆ{π₯ β π« π β£ 2 β€ (β―βπ₯)}) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 βwex 1781 β wcel 2106 β wne 2940 {crab 3432 Vcvv 3474 β c0 4322 π« cpw 4602 {csn 4628 class class class wbr 5148 dom cdm 5676 βΆwf 6539 βcfv 6543 0cc0 11112 1c1 11113 β€ cle 11251 2c2 12269 β―chash 14292 Vtxcvtx 28294 iEdgciedg 28295 Edgcedg 28345 UHGraphcuhgr 28354 Cyclesccycls 29080 AcyclicGraphcacycgr 34202 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-n0 12475 df-xnn0 12547 df-z 12561 df-uz 12825 df-fz 13487 df-fzo 13630 df-hash 14293 df-word 14467 df-concat 14523 df-s1 14548 df-s2 14801 df-edg 28346 df-uhgr 28356 df-wlks 28894 df-wlkson 28895 df-trls 28987 df-trlson 28988 df-pths 29011 df-pthson 29013 df-cycls 29082 df-acycgr 34203 |
This theorem is referenced by: upgracycumgr 34213 |
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