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Theorem uvtxnbgrb 29433
Description: A vertex is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 14-Feb-2022.)
Hypothesis
Ref Expression
uvtxnbgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
uvtxnbgrb (𝑁𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))

Proof of Theorem uvtxnbgrb
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 uvtxnbgr.v . . 3 𝑉 = (Vtx‘𝐺)
21uvtxnbgr 29432 . 2 (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
3 simpl 482 . . . 4 ((𝑁𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁𝑉)
4 raleleq 3340 . . . . . 6 ((𝑉 ∖ {𝑁}) = (𝐺 NeighbVtx 𝑁) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
54eqcoms 2743 . . . . 5 ((𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
65adantl 481 . . . 4 ((𝑁𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
71uvtxel 29420 . . . 4 (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
83, 6, 7sylanbrc 583 . . 3 ((𝑁𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁 ∈ (UnivVtx‘𝐺))
98ex 412 . 2 (𝑁𝑉 → ((𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}) → 𝑁 ∈ (UnivVtx‘𝐺)))
102, 9impbid2 226 1 (𝑁𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cdif 3960  {csn 4631  cfv 6563  (class class class)co 7431  Vtxcvtx 29028   NeighbVtx cnbgr 29364  UnivVtxcuvtx 29417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-nbgr 29365  df-uvtx 29418
This theorem is referenced by:  nbusgrvtxm1uvtx  29437  uvtxupgrres  29440  nbcplgr  29466
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