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Mirrors > Home > MPE Home > Th. List > vdgn0frgrv2 | Structured version Visualization version GIF version |
Description: A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.) |
Ref | Expression |
---|---|
vdn1frgrv2.v | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
vdgn0frgrv2 | β’ ((πΊ β FriendGraph β§ π β π) β (1 < (β―βπ) β ((VtxDegβπΊ)βπ) β 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrconngr 29241 | . . 3 β’ (πΊ β FriendGraph β πΊ β ConnGraph) | |
2 | frgrusgr 29208 | . . . 4 β’ (πΊ β FriendGraph β πΊ β USGraph) | |
3 | usgrumgr 28133 | . . . 4 β’ (πΊ β USGraph β πΊ β UMGraph) | |
4 | 2, 3 | syl 17 | . . 3 β’ (πΊ β FriendGraph β πΊ β UMGraph) |
5 | vdn1frgrv2.v | . . . . 5 β’ π = (VtxβπΊ) | |
6 | 5 | vdn0conngrumgrv2 29143 | . . . 4 β’ (((πΊ β ConnGraph β§ πΊ β UMGraph) β§ (π β π β§ 1 < (β―βπ))) β ((VtxDegβπΊ)βπ) β 0) |
7 | 6 | ex 414 | . . 3 β’ ((πΊ β ConnGraph β§ πΊ β UMGraph) β ((π β π β§ 1 < (β―βπ)) β ((VtxDegβπΊ)βπ) β 0)) |
8 | 1, 4, 7 | syl2anc 585 | . 2 β’ (πΊ β FriendGraph β ((π β π β§ 1 < (β―βπ)) β ((VtxDegβπΊ)βπ) β 0)) |
9 | 8 | expdimp 454 | 1 β’ ((πΊ β FriendGraph β§ π β π) β (1 < (β―βπ) β ((VtxDegβπΊ)βπ) β 0)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 class class class wbr 5106 βcfv 6497 0cc0 11052 1c1 11053 < clt 11190 β―chash 14231 Vtxcvtx 27950 UMGraphcumgr 28035 USGraphcusgr 28103 VtxDegcvtxdg 28416 ConnGraphcconngr 29133 FriendGraph cfrgr 29205 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 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-tp 4592 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-oadd 8417 df-er 8649 df-map 8768 df-pm 8769 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-dju 9838 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-xnn0 12487 df-z 12501 df-uz 12765 df-xadd 13035 df-fz 13426 df-fzo 13569 df-hash 14232 df-word 14404 df-concat 14460 df-s1 14485 df-s2 14738 df-s3 14739 df-edg 28002 df-uhgr 28012 df-upgr 28036 df-umgr 28037 df-uspgr 28104 df-usgr 28105 df-vtxdg 28417 df-wlks 28550 df-wlkson 28551 df-trls 28643 df-trlson 28644 df-pths 28667 df-spths 28668 df-pthson 28669 df-spthson 28670 df-conngr 29134 df-frgr 29206 |
This theorem is referenced by: vdgfrgrgt2 29245 frgrregord013 29342 |
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