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| Mirrors > Home > MPE Home > Th. List > uhgr0edg0rgrb | Structured version Visualization version GIF version | ||
| Description: A hypergraph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.) |
| Ref | Expression |
|---|---|
| uhgr0edg0rgrb | β’ (πΊ β UHGraph β (πΊ RegGraph 0 β (EdgβπΊ) = β )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2731 | . . . . . 6 β’ (VtxβπΊ) = (VtxβπΊ) | |
| 2 | eqid 2731 | . . . . . 6 β’ (EdgβπΊ) = (EdgβπΊ) | |
| 3 | 1, 2 | uhgrvd00 29055 | . . . . 5 β’ (πΊ β UHGraph β (βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = 0 β (EdgβπΊ) = β )) |
| 4 | 3 | com12 32 | . . . 4 β’ (βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = 0 β (πΊ β UHGraph β (EdgβπΊ) = β )) |
| 5 | 4 | adantl 481 | . . 3 β’ ((0 β β0* β§ βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = 0) β (πΊ β UHGraph β (EdgβπΊ) = β )) |
| 6 | eqid 2731 | . . . 4 β’ (VtxDegβπΊ) = (VtxDegβπΊ) | |
| 7 | 1, 6 | rgrprop 29081 | . . 3 β’ (πΊ RegGraph 0 β (0 β β0* β§ βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = 0)) |
| 8 | 5, 7 | syl11 33 | . 2 β’ (πΊ β UHGraph β (πΊ RegGraph 0 β (EdgβπΊ) = β )) |
| 9 | uhgr0edg0rgr 29094 | . . 3 β’ ((πΊ β UHGraph β§ (EdgβπΊ) = β ) β πΊ RegGraph 0) | |
| 10 | 9 | ex 412 | . 2 β’ (πΊ β UHGraph β ((EdgβπΊ) = β β πΊ RegGraph 0)) |
| 11 | 8, 10 | impbid 211 | 1 β’ (πΊ β UHGraph β (πΊ RegGraph 0 β (EdgβπΊ) = β )) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 βwral 3060 β c0 4323 class class class wbr 5149 βcfv 6544 0cc0 11113 β0*cxnn0 12549 Vtxcvtx 28520 Edgcedg 28571 UHGraphcuhgr 28580 VtxDegcvtxdg 28986 RegGraph crgr 29076 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-xadd 13098 df-fz 13490 df-hash 14296 df-edg 28572 df-uhgr 28582 df-vtxdg 28987 df-rgr 29078 |
| This theorem is referenced by: usgr0edg0rusgr 29096 |
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