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Mirrors > Home > MPE Home > Th. List > 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 | uvtxnbgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | nbgrssovtx 27146 | . . 3 ⊢ (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}) |
3 | 2 | a1i 11 | . 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁})) |
4 | 1 | uvtxnbgrss 27177 | . 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑉 ∖ {𝑁}) ⊆ (𝐺 NeighbVtx 𝑁)) |
5 | 3, 4 | eqssd 3987 | 1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∖ cdif 3936 ⊆ wss 3939 {csn 4570 ‘cfv 6358 (class class class)co 7159 Vtxcvtx 26784 NeighbVtx cnbgr 27117 UnivVtxcuvtx 27170 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-nbgr 27118 df-uvtx 27171 |
This theorem is referenced by: uvtxnbgrb 27186 uvtxnm1nbgr 27189 uvtxupgrres 27193 |
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