Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uvtxnbgr | Structured version Visualization version GIF version |
Description: A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 23-Mar-2021.) |
Ref | Expression |
---|---|
uvtxnbgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
uvtxnbgr | ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6131 | . . 3 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → 𝐺 ∈ V) | |
2 | uvtxnbgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | nbgrssovtx 40586 | . . 3 ⊢ (𝐺 ∈ V → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁})) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁})) |
5 | 2 | uvtxanbgr 40618 | . 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑉 ∖ {𝑁}) ⊆ (𝐺 NeighbVtx 𝑁)) |
6 | 4, 5 | eqssd 3585 | 1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 ‘cfv 5804 (class class class)co 6549 Vtxcvtx 25673 NeighbVtx cnbgr 40550 UnivVtxcuvtxa 40551 |
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 |
This theorem is referenced by: uvtxnbgrb 40628 uvtxanm1nbgr 40631 uvtxupgrres 40635 |
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