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Theorem nbcplgr 40656
 Description: In a complete graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.)
Hypothesis
Ref Expression
nbcplgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbcplgr ((𝐺 ∈ ComplGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))

Proof of Theorem nbcplgr
StepHypRef Expression
1 nbcplgr.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
21cplgruvtxb 40637 . . . . . 6 (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
32ibi 255 . . . . 5 (𝐺 ∈ ComplGraph → (UnivVtx‘𝐺) = 𝑉)
43eqcomd 2616 . . . 4 (𝐺 ∈ ComplGraph → 𝑉 = (UnivVtx‘𝐺))
54eleq2d 2673 . . 3 (𝐺 ∈ ComplGraph → (𝑁𝑉𝑁 ∈ (UnivVtx‘𝐺)))
65biimpa 500 . 2 ((𝐺 ∈ ComplGraph ∧ 𝑁𝑉) → 𝑁 ∈ (UnivVtx‘𝐺))
71uvtxnbgrb 40628 . . 3 (𝑁𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))
87adantl 481 . 2 ((𝐺 ∈ ComplGraph ∧ 𝑁𝑉) → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))
96, 8mpbid 221 1 ((𝐺 ∈ ComplGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  {csn 4125  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   NeighbVtx cnbgr 40550  UnivVtxcuvtxa 40551  ComplGraphccplgr 40552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-nbgr 40554  df-uvtxa 40556  df-cplgr 40557 This theorem is referenced by:  cusgrsizeindslem  40667  cusgrrusgr  40781
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