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| Mirrors > Home > MPE Home > Th. List > 0edg0rgr | Structured version Visualization version GIF version | ||
| Description: A graph is 0-regular if it has no edges. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
| Ref | Expression |
|---|---|
| 0edg0rgr | β’ ((πΊ β π β§ (iEdgβπΊ) = β ) β πΊ RegGraph 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpr 485 | . . . . 5 β’ (((πΊ β π β§ (iEdgβπΊ) = β ) β§ π£ β (VtxβπΊ)) β π£ β (VtxβπΊ)) | |
| 2 | simplr 767 | . . . . 5 β’ (((πΊ β π β§ (iEdgβπΊ) = β ) β§ π£ β (VtxβπΊ)) β (iEdgβπΊ) = β ) | |
| 3 | eqid 2732 | . . . . . 6 β’ (VtxβπΊ) = (VtxβπΊ) | |
| 4 | eqid 2732 | . . . . . 6 β’ (iEdgβπΊ) = (iEdgβπΊ) | |
| 5 | 3, 4 | vtxdg0e 28986 | . . . . 5 β’ ((π£ β (VtxβπΊ) β§ (iEdgβπΊ) = β ) β ((VtxDegβπΊ)βπ£) = 0) |
| 6 | 1, 2, 5 | syl2anc 584 | . . . 4 β’ (((πΊ β π β§ (iEdgβπΊ) = β ) β§ π£ β (VtxβπΊ)) β ((VtxDegβπΊ)βπ£) = 0) |
| 7 | 6 | ralrimiva 3146 | . . 3 β’ ((πΊ β π β§ (iEdgβπΊ) = β ) β βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = 0) |
| 8 | 0xnn0 12554 | . . 3 β’ 0 β β0* | |
| 9 | 7, 8 | jctil 520 | . 2 β’ ((πΊ β π β§ (iEdgβπΊ) = β ) β (0 β β0* β§ βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = 0)) |
| 10 | 8 | a1i 11 | . . 3 β’ ((iEdgβπΊ) = β β 0 β β0*) |
| 11 | eqid 2732 | . . . 4 β’ (VtxDegβπΊ) = (VtxDegβπΊ) | |
| 12 | 3, 11 | isrgr 29071 | . . 3 β’ ((πΊ β π β§ 0 β β0*) β (πΊ RegGraph 0 β (0 β β0* β§ βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = 0))) |
| 13 | 10, 12 | sylan2 593 | . 2 β’ ((πΊ β π β§ (iEdgβπΊ) = β ) β (πΊ RegGraph 0 β (0 β β0* β§ βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = 0))) |
| 14 | 9, 13 | mpbird 256 | 1 β’ ((πΊ β π β§ (iEdgβπΊ) = β ) β πΊ RegGraph 0) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 β c0 4322 class class class wbr 5148 βcfv 6543 0cc0 11112 β0*cxnn0 12548 Vtxcvtx 28511 iEdgciedg 28512 VtxDegcvtxdg 28977 RegGraph crgr 29067 |
| 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-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-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 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-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-xadd 13097 df-fz 13489 df-hash 14295 df-vtxdg 28978 df-rgr 29069 |
| This theorem is referenced by: uhgr0edg0rgr 29085 |
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