Theorem iscplgrnb 29396

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 29395	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
3	1	uvtxel 29368	. . . . 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 539	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → (𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
6	5	ralbidva 3154	. 2 ⊢ (𝐺 ∈ 𝑊 → (∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
7	2, 6	bitrd 279	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ∖ cdif 3895  {csn 4575  ‘cfv 6486  (class class class)co 7352  Vtxcvtx 28976   NeighbVtx cnbgr 29312  UnivVtxcuvtx 29365  ComplGraphccplgr 29389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-uvtx 29366  df-cplgr 29391

This theorem is referenced by:  iscplgredg  29397  iscusgredg  29403  cplgr3v  29415