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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 30057 | . . . 4 β’ ((πΊ β FriendGraph β§ π β π) β (1 < (β―βπ) β ((VtxDegβπΊ)βπ) β 0)) |
| 3 | 2 | imp 406 | . . 3 β’ (((πΊ β FriendGraph β§ π β π) β§ 1 < (β―βπ)) β ((VtxDegβπΊ)βπ) β 0) |
| 4 | 1 | vdgn1frgrv2 30058 | . . . 4 β’ ((πΊ β FriendGraph β§ π β π) β (1 < (β―βπ) β ((VtxDegβπΊ)βπ) β 1)) |
| 5 | 4 | imp 406 | . . 3 β’ (((πΊ β FriendGraph β§ π β π) β§ 1 < (β―βπ)) β ((VtxDegβπΊ)βπ) β 1) |
| 6 | 1 | vtxdgelxnn0 29238 | . . . . 5 β’ (π β π β ((VtxDegβπΊ)βπ) β β0*) |
| 7 | xnn0n0n1ge2b 13117 | . . . . 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 724 | . . 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 709 | . 2 β’ (((πΊ β FriendGraph β§ π β π) β§ 1 < (β―βπ)) β 2 β€ ((VtxDegβπΊ)βπ)) |
| 11 | 10 | ex 412 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6537 0cc0 11112 1c1 11113 < clt 11252 β€ cle 11253 2c2 12271 β0*cxnn0 12548 β―chash 14295 Vtxcvtx 28764 VtxDegcvtxdg 29231 FriendGraph cfrgr 30020 |
| 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 7722 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 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-rmo 3370 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-tp 4628 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 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 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-xadd 13099 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-concat 14527 df-s1 14552 df-s2 14805 df-s3 14806 df-edg 28816 df-uhgr 28826 df-upgr 28850 df-umgr 28851 df-uspgr 28918 df-usgr 28919 df-vtxdg 29232 df-wlks 29365 df-wlkson 29366 df-trls 29458 df-trlson 29459 df-pths 29482 df-spths 29483 df-pthson 29484 df-spthson 29485 df-conngr 29949 df-frgr 30021 |
| This theorem is referenced by: frgrwopreglem2 30075 |
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