MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscplgrnb Structured version   Visualization version   GIF version

Theorem iscplgrnb 26665
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 26663 . 2 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
31uvtxel 26632 . . . . 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 536 . . 3 ((𝐺𝑊𝑣𝑉) → (𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
65ralbidva 3167 . 2 (𝐺𝑊 → (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
72, 6bitrd 271 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3090  cdif 3767  {csn 4369  cfv 6102  (class class class)co 6879  Vtxcvtx 26230   NeighbVtx cnbgr 26565  UnivVtxcuvtx 26628  ComplGraphccplgr 26657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-uvtx 26629  df-cplgr 26659
This theorem is referenced by:  iscplgredg  26666  iscusgredg  26672  cplgr3v  26684
  Copyright terms: Public domain W3C validator