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Mirrors > Home > MPE Home > Th. List > 0vtxrgr | Structured version Visualization version GIF version |
Description: A null graph (with no vertices) is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
0vtxrgr | β’ ((πΊ β π β§ (VtxβπΊ) = β ) β βπ β β0* πΊ RegGraph π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . 3 β’ (((πΊ β π β§ (VtxβπΊ) = β ) β§ π β β0*) β π β β0*) | |
2 | rzal 4471 | . . . 4 β’ ((VtxβπΊ) = β β βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = π) | |
3 | 2 | ad2antlr 726 | . . 3 β’ (((πΊ β π β§ (VtxβπΊ) = β ) β§ π β β0*) β βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = π) |
4 | eqid 2737 | . . . . 5 β’ (VtxβπΊ) = (VtxβπΊ) | |
5 | eqid 2737 | . . . . 5 β’ (VtxDegβπΊ) = (VtxDegβπΊ) | |
6 | 4, 5 | isrgr 28549 | . . . 4 β’ ((πΊ β π β§ π β β0*) β (πΊ RegGraph π β (π β β0* β§ βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = π))) |
7 | 6 | adantlr 714 | . . 3 β’ (((πΊ β π β§ (VtxβπΊ) = β ) β§ π β β0*) β (πΊ RegGraph π β (π β β0* β§ βπ£ β (VtxβπΊ)((VtxDegβπΊ)βπ£) = π))) |
8 | 1, 3, 7 | mpbir2and 712 | . 2 β’ (((πΊ β π β§ (VtxβπΊ) = β ) β§ π β β0*) β πΊ RegGraph π) |
9 | 8 | ralrimiva 3144 | 1 β’ ((πΊ β π β§ (VtxβπΊ) = β ) β βπ β β0* πΊ RegGraph π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 β c0 4287 class class class wbr 5110 βcfv 6501 β0*cxnn0 12492 Vtxcvtx 27989 VtxDegcvtxdg 28455 RegGraph crgr 28545 |
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-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-iota 6453 df-fv 6509 df-rgr 28547 |
This theorem is referenced by: 0vtxrusgr 28567 |
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