Theorem 0vtxrgr 29557

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 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegGraph 𝑘)

Distinct variable groups:   𝑘,𝐺   𝑘,𝑊

Proof of Theorem 0vtxrgr

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝑘 ∈ ℕ0*)
2		rzal 4458	. . . 4 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
3	2	ad2antlr 727	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)
4		eqid 2733	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2733	. . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
6	4, 5	isrgr 29540	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑘 ∈ ℕ0*) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
7	6	adantlr 715	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 𝑘)))
8	1, 3, 7	mpbir2and 713	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺 RegGraph 𝑘)
9	8	ralrimiva 3125	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegGraph 𝑘)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∅c0 4282   class class class wbr 5093  ‘cfv 6486  ℕ₀^*cxnn0 12461  Vtxcvtx 28976  VtxDegcvtxdg 29446   RegGraph crgr 29536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-iota 6442  df-fv 6494  df-rgr 29538

This theorem is referenced by:  0vtxrusgr  29558