Theorem iscplgrnb 29671

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	⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (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

Step	Hyp	Ref	Expression
1		cplgruvtxb.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	iscplgr 29670	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
3	1	uvtxel 29643	. . . . 5 ⊢ (𝑣 ∈ (UnivVtx‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
4	3	a1i 11	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝑣 ∈ (UnivVtx‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
5	4	baibd 548	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
6	5	ralbidva 3186	. 2 ⊢ (𝐺 ∈ 𝑊 → (∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
7	2, 6	bitrd 282	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ∖ cdif 3904  {csn 4585  ‘cfv 6525  (class class class)co 7400  Vtxcvtx 29251   NeighbVtx cnbgr 29587  UnivVtxcuvtx 29640  ComplGraphccplgr 29664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-uvtx 29641  df-cplgr 29666

This theorem is referenced by:  iscplgredg  29672  iscusgredg  29678  cplgr3v  29690