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Mirrors > Home > MPE Home > Th. List > 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 28069 | . . . . . 6 ⊢ (𝐺 ∈ ComplGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉)) |
3 | 2 | ibi 266 | . . . . 5 ⊢ (𝐺 ∈ ComplGraph → (UnivVtx‘𝐺) = 𝑉) |
4 | 3 | eqcomd 2742 | . . . 4 ⊢ (𝐺 ∈ ComplGraph → 𝑉 = (UnivVtx‘𝐺)) |
5 | 4 | eleq2d 2822 | . . 3 ⊢ (𝐺 ∈ ComplGraph → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (UnivVtx‘𝐺))) |
6 | 5 | biimpa 477 | . 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (UnivVtx‘𝐺)) |
7 | 1 | uvtxnbgrb 28057 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))) |
8 | 7 | adantl 482 | . 2 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))) |
9 | 6, 8 | mpbid 231 | 1 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∖ cdif 3895 {csn 4573 ‘cfv 6479 (class class class)co 7337 Vtxcvtx 27655 NeighbVtx cnbgr 27988 UnivVtxcuvtx 28041 ComplGraphccplgr 28065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-nbgr 27989 df-uvtx 28042 df-cplgr 28067 |
This theorem is referenced by: cusgrsizeindslem 28107 cusgrrusgr 28237 |
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