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Theorem iscplgrnb 29448
Description: A graph is complete iff all vertices are neighbors of all vertices. (Contributed by AV, 1-Nov-2020.)
Hypothesis
Ref Expression
cplgruvtxb.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
iscplgrnb (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
Distinct variable groups:   𝑣,𝐺   𝑣,𝑉   𝑛,𝐺,𝑣   𝑛,𝑉   𝑣,𝑊
Allowed substitution hint:   𝑊(𝑛)

Proof of Theorem iscplgrnb
StepHypRef Expression
1 cplgruvtxb.v . . 3 𝑉 = (Vtx‘𝐺)
21iscplgr 29447 . 2 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
31uvtxel 29420 . . . . 5 (𝑣 ∈ (UnivVtx‘𝐺) ↔ (𝑣𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
43a1i 11 . . . 4 (𝐺𝑊 → (𝑣 ∈ (UnivVtx‘𝐺) ↔ (𝑣𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
54baibd 539 . . 3 ((𝐺𝑊𝑣𝑉) → (𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
65ralbidva 3174 . 2 (𝐺𝑊 → (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
72, 6bitrd 279 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cdif 3960  {csn 4631  cfv 6563  (class class class)co 7431  Vtxcvtx 29028   NeighbVtx cnbgr 29364  UnivVtxcuvtx 29417  ComplGraphccplgr 29441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-uvtx 29418  df-cplgr 29443
This theorem is referenced by:  iscplgredg  29449  iscusgredg  29455  cplgr3v  29467
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