Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nbcplgr | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nbcplgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
nbcplgr | ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbcplgr.v | . . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | cplgruvtxb 40637 | . . . . . 6 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
3 | 2 | ibi 255 | . . . . 5 ⊢ (𝐺 ∈ ComplGraph → (UnivVtx‘𝐺) = 𝑉) |
4 | 3 | eqcomd 2616 | . . . 4 ⊢ (𝐺 ∈ ComplGraph → 𝑉 = (UnivVtx‘𝐺)) |
5 | 4 | eleq2d 2673 | . . 3 ⊢ (𝐺 ∈ ComplGraph → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (UnivVtx‘𝐺))) |
6 | 5 | biimpa 500 | . 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (UnivVtx‘𝐺)) |
7 | 1 | uvtxnbgrb 40628 | . . 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 221 | 1 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (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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