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Theorem 0vtxrusgr 40777
 Description: A graph with no vertices and an empty edge function is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Assertion
Ref Expression
0vtxrusgr ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘)
Distinct variable groups:   𝑘,𝐺   𝑘,𝑊

Proof of Theorem 0vtxrusgr
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 usgr0v 40467 . . . 4 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph )
21adantr 480 . . 3 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺 ∈ USGraph )
3 0vtxrgr 40776 . . . . . 6 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑣 ∈ ℕ0* 𝐺 RegGraph 𝑣)
4 breq2 4587 . . . . . . . 8 (𝑣 = 𝑘 → (𝐺 RegGraph 𝑣𝐺 RegGraph 𝑘))
54rspccva 3281 . . . . . . 7 ((∀𝑣 ∈ ℕ0* 𝐺 RegGraph 𝑣𝑘 ∈ ℕ0*) → 𝐺 RegGraph 𝑘)
65ex 449 . . . . . 6 (∀𝑣 ∈ ℕ0* 𝐺 RegGraph 𝑣 → (𝑘 ∈ ℕ0*𝐺 RegGraph 𝑘))
73, 6syl 17 . . . . 5 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝑘 ∈ ℕ0*𝐺 RegGraph 𝑘))
873adant3 1074 . . . 4 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → (𝑘 ∈ ℕ0*𝐺 RegGraph 𝑘))
98imp 444 . . 3 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺 RegGraph 𝑘)
10 isrusgr 40761 . . . 4 ((𝐺𝑊𝑘 ∈ ℕ0*) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
11103ad2antl1 1216 . . 3 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘)))
122, 9, 11mpbir2and 959 . 2 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺 RegUSGraph 𝑘)
1312ralrimiva 2949 1 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  ℕ0*cxnn0 11240  Vtxcvtx 25673  iEdgciedg 25674   USGraph cusgr 40379   RegGraph crgr 40755   RegUSGraph crusgr 40756 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-8 1979  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-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-2 10956  df-uhgr 25724  df-upgr 25749  df-uspgr 40380  df-usgr 40381  df-rgr 40757  df-rusgr 40758 This theorem is referenced by:  0uhgrrusgr  40778  0grrusgr  40779
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