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| Mirrors > Home > MPE Home > Th. List > vdgfrgrgt2 | Structured version Visualization version GIF version | ||
| Description: Any vertex in a friendship graph (with more than one vertex - then, actually, the graph must have at least three vertices, because otherwise, it would not be a friendship graph) has at least degree 2, see remark 3 in [MertziosUnger] p. 153 (after Proposition 1): "It follows that deg(v) >= 2 for every node v of a friendship graph". (Contributed by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 5-Apr-2021.) |
| Ref | Expression |
|---|---|
| vdn1frgrv2.v | β’ π = (VtxβπΊ) |
| Ref | Expression |
|---|---|
| vdgfrgrgt2 | β’ ((πΊ β FriendGraph β§ π β π) β (1 < (β―βπ) β 2 β€ ((VtxDegβπΊ)βπ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vdn1frgrv2.v | . . . . 5 β’ π = (VtxβπΊ) | |
| 2 | 1 | vdgn0frgrv2 29537 | . . . 4 β’ ((πΊ β FriendGraph β§ π β π) β (1 < (β―βπ) β ((VtxDegβπΊ)βπ) β 0)) |
| 3 | 2 | imp 407 | . . 3 β’ (((πΊ β FriendGraph β§ π β π) β§ 1 < (β―βπ)) β ((VtxDegβπΊ)βπ) β 0) |
| 4 | 1 | vdgn1frgrv2 29538 | . . . 4 β’ ((πΊ β FriendGraph β§ π β π) β (1 < (β―βπ) β ((VtxDegβπΊ)βπ) β 1)) |
| 5 | 4 | imp 407 | . . 3 β’ (((πΊ β FriendGraph β§ π β π) β§ 1 < (β―βπ)) β ((VtxDegβπΊ)βπ) β 1) |
| 6 | 1 | vtxdgelxnn0 28718 | . . . . 5 β’ (π β π β ((VtxDegβπΊ)βπ) β β0*) |
| 7 | xnn0n0n1ge2b 13107 | . . . . 5 β’ (((VtxDegβπΊ)βπ) β β0* β ((((VtxDegβπΊ)βπ) β 0 β§ ((VtxDegβπΊ)βπ) β 1) β 2 β€ ((VtxDegβπΊ)βπ))) | |
| 8 | 6, 7 | syl 17 | . . . 4 β’ (π β π β ((((VtxDegβπΊ)βπ) β 0 β§ ((VtxDegβπΊ)βπ) β 1) β 2 β€ ((VtxDegβπΊ)βπ))) |
| 9 | 8 | ad2antlr 725 | . . 3 β’ (((πΊ β FriendGraph β§ π β π) β§ 1 < (β―βπ)) β ((((VtxDegβπΊ)βπ) β 0 β§ ((VtxDegβπΊ)βπ) β 1) β 2 β€ ((VtxDegβπΊ)βπ))) |
| 10 | 3, 5, 9 | mpbi2and 710 | . 2 β’ (((πΊ β FriendGraph β§ π β π) β§ 1 < (β―βπ)) β 2 β€ ((VtxDegβπΊ)βπ)) |
| 11 | 10 | ex 413 | 1 β’ ((πΊ β FriendGraph β§ π β π) β (1 < (β―βπ) β 2 β€ ((VtxDegβπΊ)βπ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 β wne 2940 class class class wbr 5147 βcfv 6540 0cc0 11106 1c1 11107 < clt 11244 β€ cle 11245 2c2 12263 β0*cxnn0 12540 β―chash 14286 Vtxcvtx 28245 VtxDegcvtxdg 28711 FriendGraph cfrgr 29500 |
| 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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
| 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-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-xadd 13089 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-s3 14796 df-edg 28297 df-uhgr 28307 df-upgr 28331 df-umgr 28332 df-uspgr 28399 df-usgr 28400 df-vtxdg 28712 df-wlks 28845 df-wlkson 28846 df-trls 28938 df-trlson 28939 df-pths 28962 df-spths 28963 df-pthson 28964 df-spthson 28965 df-conngr 29429 df-frgr 29501 |
| This theorem is referenced by: frgrwopreglem2 29555 |
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