Theorem nbcplgr 29844

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 ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))

Proof of Theorem nbcplgr

Step	Hyp	Ref	Expression
1		nbcplgr.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	cplgruvtxb 29823	. . . . . 6 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
3	2	ibi 270	. . . . 5 ⊢ (𝐺 ∈ ComplGraph → (UnivVtx‘𝐺) = 𝑉)
4	3	eqcomd 2771	. . . 4 ⊢ (𝐺 ∈ ComplGraph → 𝑉 = (UnivVtx‘𝐺))
5	4	eleq2d 2851	. . 3 ⊢ (𝐺 ∈ ComplGraph → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (UnivVtx‘𝐺)))
6	5	biimpa 482	. 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (UnivVtx‘𝐺))
7	1	uvtxnbgrb 29811	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))
8	7	adantl 487	. 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))
9	6, 8	mpbid 235	1 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∖ cdif 3903  {csn 4591  ‘cfv 6540  (class class class)co 7419  Vtxcvtx 29403   NeighbVtx cnbgr 29742  UnivVtxcuvtx 29795  ComplGraphccplgr 29819

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-nbgr 29743  df-uvtx 29796  df-cplgr 29821

This theorem is used by:  cusgrsizeindslem  29861  cusgrrusgr  29991