nbcplgr - Metamath Proof Explorer

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Theorem nbcplgr 29361

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 29340	. . . . . 6 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
3	2	ibi 267	. . . . 5 ⊢ (𝐺 ∈ ComplGraph → (UnivVtx‘𝐺) = 𝑉)
4	3	eqcomd 2735	. . . 4 ⊢ (𝐺 ∈ ComplGraph → 𝑉 = (UnivVtx‘𝐺))
5	4	eleq2d 2814	. . 3 ⊢ (𝐺 ∈ ComplGraph → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (UnivVtx‘𝐺)))
6	5	biimpa 476	. 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (UnivVtx‘𝐺))
7	1	uvtxnbgrb 29328	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))
8	7	adantl 481	. 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))
9	6, 8	mpbid 232	1 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∖ cdif 3911 {csn 4589 ‘cfv 6511 (class class class)co 7387 Vtxcvtx 28923 NeighbVtx cnbgr 29259 UnivVtxcuvtx 29312 ComplGraphccplgr 29336

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-nbgr 29260 df-uvtx 29313 df-cplgr 29338

This theorem is referenced by: cusgrsizeindslem 29379 cusgrrusgr 29509

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