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Theorem 0vtxrgr 28833
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 4509 . . . 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 2733 . . . . 5 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
5 eqid 2733 . . . . 5 (VtxDegβ€˜πΊ) = (VtxDegβ€˜πΊ)
64, 5isrgr 28816 . . . 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 3147 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 3062  βˆ…c0 4323   class class class wbr 5149  β€˜cfv 6544  β„•0*cxnn0 12544  Vtxcvtx 28256  VtxDegcvtxdg 28722   RegGraph crgr 28812
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6496  df-fv 6552  df-rgr 28814
This theorem is referenced by:  0vtxrusgr  28834
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