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Theorem dfnbgr2 27046
Description: Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020.) (Revised by AV, 21-Mar-2021.)
Hypotheses
Ref Expression
nbgrval.v 𝑉 = (Vtx‘𝐺)
nbgrval.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
dfnbgr2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺,𝑛   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛
Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem dfnbgr2
StepHypRef Expression
1 nbgrval.v . . 3 𝑉 = (Vtx‘𝐺)
2 nbgrval.e . . 3 𝐸 = (Edg‘𝐺)
31, 2nbgrval 27045 . 2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
4 prssg 4744 . . . . . 6 ((𝑁𝑉𝑛 ∈ V) → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
54elvd 3498 . . . . 5 (𝑁𝑉 → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
65bicomd 224 . . . 4 (𝑁𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁𝑒𝑛𝑒)))
76rexbidv 3294 . . 3 (𝑁𝑉 → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)))
87rabbidv 3478 . 2 (𝑁𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
93, 8eqtrd 2853 1 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  {crab 3139  Vcvv 3492  cdif 3930  wss 3933  {csn 4557  {cpr 4559  cfv 6348  (class class class)co 7145  Vtxcvtx 26708  Edgcedg 26759   NeighbVtx cnbgr 27041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-nbgr 27042
This theorem is referenced by: (None)
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