nbcplgr - Metamath Proof Explorer

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Theorem nbcplgr 29523

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 29502	. . . . . 6 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
3	2	ibi 267	. . . . 5 ⊢ (𝐺 ∈ ComplGraph → (UnivVtx‘𝐺) = 𝑉)
4	3	eqcomd 2743	. . . 4 ⊢ (𝐺 ∈ ComplGraph → 𝑉 = (UnivVtx‘𝐺))
5	4	eleq2d 2823	. . 3 ⊢ (𝐺 ∈ ComplGraph → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (UnivVtx‘𝐺)))
6	5	biimpa 476	. 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (UnivVtx‘𝐺))
7	1	uvtxnbgrb 29490	. . 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 1542 ∈ wcel 2114 ∖ cdif 3900 {csn 4582 ‘cfv 6500 (class class class)co 7368 Vtxcvtx 29085 NeighbVtx cnbgr 29421 UnivVtxcuvtx 29474 ComplGraphccplgr 29498

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-nbgr 29422 df-uvtx 29475 df-cplgr 29500

This theorem is referenced by: cusgrsizeindslem 29541 cusgrrusgr 29671

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