Theorem 0vgrargra 26464

Description: A graph with no vertices is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.)

Assertion

Ref	Expression
0vgrargra	⊢ (𝐸 ∈ 𝑉 → ∀𝑘 ∈ ℕ0 ⟨∅, 𝐸⟩ RegGrph 𝑘)

Distinct variable groups:   𝑘,𝐸   𝑘,𝑉

Proof of Theorem 0vgrargra

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 476	. . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
2		ral0 4028	. . . 4 ⊢ ∀𝑝 ∈ ∅ ((∅ VDeg 𝐸)‘𝑝) = 𝑘
3	2	a1i 11	. . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑘 ∈ ℕ0) → ∀𝑝 ∈ ∅ ((∅ VDeg 𝐸)‘𝑝) = 𝑘)
4		0ex 4718	. . . 4 ⊢ ∅ ∈ V
5		isrgra 26453	. . . 4 ⊢ ((∅ ∈ V ∧ 𝐸 ∈ 𝑉 ∧ 𝑘 ∈ ℕ0) → (⟨∅, 𝐸⟩ RegGrph 𝑘 ↔ (𝑘 ∈ ℕ0 ∧ ∀𝑝 ∈ ∅ ((∅ VDeg 𝐸)‘𝑝) = 𝑘)))
6	4, 5	mp3an1 1403	. . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑘 ∈ ℕ0) → (⟨∅, 𝐸⟩ RegGrph 𝑘 ↔ (𝑘 ∈ ℕ0 ∧ ∀𝑝 ∈ ∅ ((∅ VDeg 𝐸)‘𝑝) = 𝑘)))
7	1, 3, 6	mpbir2and 959	. 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑘 ∈ ℕ0) → ⟨∅, 𝐸⟩ RegGrph 𝑘)
8	7	ralrimiva 2949	1 ⊢ (𝐸 ∈ 𝑉 → ∀𝑘 ∈ ℕ0 ⟨∅, 𝐸⟩ RegGrph 𝑘)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∅c0 3874  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℕ₀cn0 11169   VDeg cvdg 26420   RegGrph crgra 26449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-rgra 26451

This theorem is referenced by: (None)