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Theorem 0vtxrgr 29088
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 485 . . 3 (((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) ∧ π‘˜ ∈ β„•0*) β†’ π‘˜ ∈ β„•0*)
2 rzal 4508 . . . 4 ((Vtxβ€˜πΊ) = βˆ… β†’ βˆ€π‘£ ∈ (Vtxβ€˜πΊ)((VtxDegβ€˜πΊ)β€˜π‘£) = π‘˜)
32ad2antlr 725 . . 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 2732 . . . . 5 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
5 eqid 2732 . . . . 5 (VtxDegβ€˜πΊ) = (VtxDegβ€˜πΊ)
64, 5isrgr 29071 . . . 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 713 . . 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 711 . 2 (((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) ∧ π‘˜ ∈ β„•0*) β†’ 𝐺 RegGraph π‘˜)
98ralrimiva 3146 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆ…c0 4322   class class class wbr 5148  β€˜cfv 6543  β„•0*cxnn0 12548  Vtxcvtx 28511  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-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-rgr 29069
This theorem is referenced by:  0vtxrusgr  29089
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