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Theorem iscplgrnb 27125
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 27124 . 2 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
31uvtxel 27097 . . . . 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 540 . . 3 ((𝐺𝑊𝑣𝑉) → (𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
65ralbidva 3193 . 2 (𝐺𝑊 → (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
72, 6bitrd 280 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  cdif 3930  {csn 4557  cfv 6348  (class class class)co 7145  Vtxcvtx 26708   NeighbVtx cnbgr 27041  UnivVtxcuvtx 27094  ComplGraphccplgr 27118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-uvtx 27095  df-cplgr 27120
This theorem is referenced by:  iscplgredg  27126  iscusgredg  27132  cplgr3v  27144
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