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Theorem 0vgrargra 26257
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 (𝐸𝑉 → ∀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.
StepHypRef Expression
1 simpr 475 . . 3 ((𝐸𝑉𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
2 ral0 4027 . . . 4 𝑝 ∈ ∅ ((∅ VDeg 𝐸)‘𝑝) = 𝑘
32a1i 11 . . 3 ((𝐸𝑉𝑘 ∈ ℕ0) → ∀𝑝 ∈ ∅ ((∅ VDeg 𝐸)‘𝑝) = 𝑘)
4 0ex 4712 . . . 4 ∅ ∈ V
5 isrgra 26246 . . . 4 ((∅ ∈ V ∧ 𝐸𝑉𝑘 ∈ ℕ0) → (⟨∅, 𝐸⟩ RegGrph 𝑘 ↔ (𝑘 ∈ ℕ0 ∧ ∀𝑝 ∈ ∅ ((∅ VDeg 𝐸)‘𝑝) = 𝑘)))
64, 5mp3an1 1402 . . 3 ((𝐸𝑉𝑘 ∈ ℕ0) → (⟨∅, 𝐸⟩ RegGrph 𝑘 ↔ (𝑘 ∈ ℕ0 ∧ ∀𝑝 ∈ ∅ ((∅ VDeg 𝐸)‘𝑝) = 𝑘)))
71, 3, 6mpbir2and 958 . 2 ((𝐸𝑉𝑘 ∈ ℕ0) → ⟨∅, 𝐸⟩ RegGrph 𝑘)
87ralrimiva 2948 1 (𝐸𝑉 → ∀𝑘 ∈ ℕ0 ⟨∅, 𝐸⟩ RegGrph 𝑘)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  c0 3873  cop 4130   class class class wbr 4577  cfv 5789  (class class class)co 6526  0cn0 11141   VDeg cvdg 26213   RegGrph crgra 26242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5753  df-fv 5797  df-ov 6529  df-oprab 6530  df-rgra 26244
This theorem is referenced by: (None)
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