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Theorem 0vtxrgr 26528
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 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺RegGraph𝑘)
Distinct variable groups:   𝑘,𝐺   𝑘,𝑊

Proof of Theorem 0vtxrgr
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . 3 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝑘 ∈ ℕ0*)
2 rzal 4106 . . . 4 ((Vtx‘𝐺) = ∅ → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
32ad2antlr 763 . . 3 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
4 eqid 2651 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2651 . . . . 5 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
64, 5isrgr 26511 . . . 4 ((𝐺𝑊𝑘 ∈ ℕ0*) → (𝐺RegGraph𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
76adantlr 751 . . 3 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → (𝐺RegGraph𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
81, 3, 7mpbir2and 977 . 2 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺RegGraph𝑘)
98ralrimiva 2995 1 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺RegGraph𝑘)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  c0 3948   class class class wbr 4685  cfv 5926  0*cxnn0 11401  Vtxcvtx 25919  VtxDegcvtxdg 26417  RegGraphcrgr 26507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-iota 5889  df-fv 5934  df-rgr 26509
This theorem is referenced by:  0vtxrusgr  26529
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