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Theorem 0vtxrusgr 26683
 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* 𝐺RegUSGraph𝑘)
Distinct variable groups:   𝑘,𝐺   𝑘,𝑊

Proof of Theorem 0vtxrusgr
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 usgr0v 26332 . . . 4 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph)
21adantr 472 . . 3 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺 ∈ USGraph)
3 0vtxrgr 26682 . . . . . 6 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑣 ∈ ℕ0* 𝐺RegGraph𝑣)
4 breq2 4808 . . . . . . . 8 (𝑣 = 𝑘 → (𝐺RegGraph𝑣𝐺RegGraph𝑘))
54rspccva 3448 . . . . . . 7 ((∀𝑣 ∈ ℕ0* 𝐺RegGraph𝑣𝑘 ∈ ℕ0*) → 𝐺RegGraph𝑘)
65ex 449 . . . . . 6 (∀𝑣 ∈ ℕ0* 𝐺RegGraph𝑣 → (𝑘 ∈ ℕ0*𝐺RegGraph𝑘))
73, 6syl 17 . . . . 5 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝑘 ∈ ℕ0*𝐺RegGraph𝑘))
873adant3 1127 . . . 4 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → (𝑘 ∈ ℕ0*𝐺RegGraph𝑘))
98imp 444 . . 3 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺RegGraph𝑘)
10 isrusgr 26667 . . . 4 ((𝐺𝑊𝑘 ∈ ℕ0*) → (𝐺RegUSGraph𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝑘)))
11103ad2antl1 1201 . . 3 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → (𝐺RegUSGraph𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝑘)))
122, 9, 11mpbir2and 995 . 2 (((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) ∧ 𝑘 ∈ ℕ0*) → 𝐺RegUSGraph𝑘)
1312ralrimiva 3104 1 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺RegUSGraph𝑘)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∅c0 4058   class class class wbr 4804  ‘cfv 6049  ℕ0*cxnn0 11555  Vtxcvtx 26073  iEdgciedg 26074  USGraphcusgr 26243  RegGraphcrgr 26661  RegUSGraphcrusgr 26662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-i2m1 10196  ax-1ne0 10197  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-2 11271  df-uhgr 26152  df-upgr 26176  df-uspgr 26244  df-usgr 26245  df-rgr 26663  df-rusgr 26664 This theorem is referenced by:  0uhgrrusgr  26684  0grrusgr  26685
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