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Theorem 0vtxrgr 28566
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.)
Assertion
Ref Expression
0vtxrgr ((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) β†’ βˆ€π‘˜ ∈ β„•0* 𝐺 RegGraph π‘˜)
Distinct variable groups:   π‘˜,𝐺   π‘˜,π‘Š

Proof of Theorem 0vtxrgr
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 simpr 486 . . 3 (((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) ∧ π‘˜ ∈ β„•0*) β†’ π‘˜ ∈ β„•0*)
2 rzal 4471 . . . 4 ((Vtxβ€˜πΊ) = βˆ… β†’ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = π‘˜)
32ad2antlr 726 . . 3 (((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) ∧ π‘˜ ∈ β„•0*) β†’ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = π‘˜)
4 eqid 2737 . . . . 5 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
5 eqid 2737 . . . . 5 (VtxDegβ€˜πΊ) = (VtxDegβ€˜πΊ)
64, 5isrgr 28549 . . . 4 ((𝐺 ∈ π‘Š ∧ π‘˜ ∈ β„•0*) β†’ (𝐺 RegGraph π‘˜ ↔ (π‘˜ ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = π‘˜)))
76adantlr 714 . . 3 (((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) ∧ π‘˜ ∈ β„•0*) β†’ (𝐺 RegGraph π‘˜ ↔ (π‘˜ ∈ β„•0* ∧ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = π‘˜)))
81, 3, 7mpbir2and 712 . 2 (((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) ∧ π‘˜ ∈ β„•0*) β†’ 𝐺 RegGraph π‘˜)
98ralrimiva 3144 1 ((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) β†’ βˆ€π‘˜ ∈ β„•0* 𝐺 RegGraph π‘˜)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆ…c0 4287   class class class wbr 5110  β€˜cfv 6501  β„•0*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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