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Mirrors > Home > MPE Home > Th. List > vtxduhgr0nedg | Structured version Visualization version GIF version |
Description: If a vertex in a hypergraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by AV, 15-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.) |
Ref | Expression |
---|---|
vtxdushgrfvedg.v | β’ π = (VtxβπΊ) |
vtxdushgrfvedg.e | β’ πΈ = (EdgβπΊ) |
vtxdushgrfvedg.d | β’ π· = (VtxDegβπΊ) |
Ref | Expression |
---|---|
vtxduhgr0nedg | β’ ((πΊ β UHGraph β§ π β π β§ (π·βπ) = 0) β Β¬ βπ£ β π {π, π£} β πΈ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdushgrfvedg.v | . . . . 5 β’ π = (VtxβπΊ) | |
2 | eqid 2736 | . . . . 5 β’ (iEdgβπΊ) = (iEdgβπΊ) | |
3 | vtxdushgrfvedg.d | . . . . 5 β’ π· = (VtxDegβπΊ) | |
4 | 1, 2, 3 | vtxd0nedgb 28144 | . . . 4 β’ (π β π β ((π·βπ) = 0 β Β¬ βπ β dom (iEdgβπΊ)π β ((iEdgβπΊ)βπ))) |
5 | 4 | adantl 482 | . . 3 β’ ((πΊ β UHGraph β§ π β π) β ((π·βπ) = 0 β Β¬ βπ β dom (iEdgβπΊ)π β ((iEdgβπΊ)βπ))) |
6 | vtxdushgrfvedg.e | . . . . . . . . 9 β’ πΈ = (EdgβπΊ) | |
7 | 6 | eleq2i 2828 | . . . . . . . 8 β’ ({π, π£} β πΈ β {π, π£} β (EdgβπΊ)) |
8 | 2 | uhgredgiedgb 27785 | . . . . . . . 8 β’ (πΊ β UHGraph β ({π, π£} β (EdgβπΊ) β βπ β dom (iEdgβπΊ){π, π£} = ((iEdgβπΊ)βπ))) |
9 | 7, 8 | bitrid 282 | . . . . . . 7 β’ (πΊ β UHGraph β ({π, π£} β πΈ β βπ β dom (iEdgβπΊ){π, π£} = ((iEdgβπΊ)βπ))) |
10 | 9 | adantr 481 | . . . . . 6 β’ ((πΊ β UHGraph β§ π β π) β ({π, π£} β πΈ β βπ β dom (iEdgβπΊ){π, π£} = ((iEdgβπΊ)βπ))) |
11 | prid1g 4708 | . . . . . . . . 9 β’ (π β π β π β {π, π£}) | |
12 | eleq2 2825 | . . . . . . . . 9 β’ ({π, π£} = ((iEdgβπΊ)βπ) β (π β {π, π£} β π β ((iEdgβπΊ)βπ))) | |
13 | 11, 12 | syl5ibcom 244 | . . . . . . . 8 β’ (π β π β ({π, π£} = ((iEdgβπΊ)βπ) β π β ((iEdgβπΊ)βπ))) |
14 | 13 | adantl 482 | . . . . . . 7 β’ ((πΊ β UHGraph β§ π β π) β ({π, π£} = ((iEdgβπΊ)βπ) β π β ((iEdgβπΊ)βπ))) |
15 | 14 | reximdv 3163 | . . . . . 6 β’ ((πΊ β UHGraph β§ π β π) β (βπ β dom (iEdgβπΊ){π, π£} = ((iEdgβπΊ)βπ) β βπ β dom (iEdgβπΊ)π β ((iEdgβπΊ)βπ))) |
16 | 10, 15 | sylbid 239 | . . . . 5 β’ ((πΊ β UHGraph β§ π β π) β ({π, π£} β πΈ β βπ β dom (iEdgβπΊ)π β ((iEdgβπΊ)βπ))) |
17 | 16 | rexlimdvw 3153 | . . . 4 β’ ((πΊ β UHGraph β§ π β π) β (βπ£ β π {π, π£} β πΈ β βπ β dom (iEdgβπΊ)π β ((iEdgβπΊ)βπ))) |
18 | 17 | con3d 152 | . . 3 β’ ((πΊ β UHGraph β§ π β π) β (Β¬ βπ β dom (iEdgβπΊ)π β ((iEdgβπΊ)βπ) β Β¬ βπ£ β π {π, π£} β πΈ)) |
19 | 5, 18 | sylbid 239 | . 2 β’ ((πΊ β UHGraph β§ π β π) β ((π·βπ) = 0 β Β¬ βπ£ β π {π, π£} β πΈ)) |
20 | 19 | 3impia 1116 | 1 β’ ((πΊ β UHGraph β§ π β π β§ (π·βπ) = 0) β Β¬ βπ£ β π {π, π£} β πΈ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 396 β§ w3a 1086 = wceq 1540 β wcel 2105 βwrex 3070 {cpr 4575 dom cdm 5620 βcfv 6479 0cc0 10972 Vtxcvtx 27655 iEdgciedg 27656 Edgcedg 27706 UHGraphcuhgr 27715 VtxDegcvtxdg 28121 |
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 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-n0 12335 df-xnn0 12407 df-z 12421 df-uz 12684 df-xadd 12950 df-fz 13341 df-hash 14146 df-edg 27707 df-uhgr 27717 df-vtxdg 28122 |
This theorem is referenced by: vtxdumgr0nedg 28149 |
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