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Theorem uvtxnbgr 26530
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.)
Hypothesis
Ref Expression
uvtxnbgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
uvtxnbgr (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))

Proof of Theorem uvtxnbgr
StepHypRef Expression
1 uvtxnbgr.v . . . 4 𝑉 = (Vtx‘𝐺)
21nbgrssovtx 26480 . . 3 (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁})
32a1i 11 . 2 (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))
41uvtxnbgrss 26519 . 2 (𝑁 ∈ (UnivVtx‘𝐺) → (𝑉 ∖ {𝑁}) ⊆ (𝐺 NeighbVtx 𝑁))
53, 4eqssd 3769 1 (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cdif 3720  wss 3723  {csn 4316  cfv 6031  (class class class)co 6793  Vtxcvtx 26095   NeighbVtx cnbgr 26447  UnivVtxcuvtx 26510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-nbgr 26448  df-uvtx 26511
This theorem is referenced by:  uvtxnbgrb  26531  uvtxnm1nbgr  26534  uvtxupgrres  26538
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