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Theorem dfnbgr2 29267
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 29266 . 2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
4 prssg 4818 . . . . . 6 ((𝑁𝑉𝑛 ∈ V) → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
54elvd 3469 . . . . 5 (𝑁𝑉 → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
65bicomd 222 . . . 4 (𝑁𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁𝑒𝑛𝑒)))
76rexbidv 3169 . . 3 (𝑁𝑉 → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)))
87rabbidv 3427 . 2 (𝑁𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
93, 8eqtrd 2766 1 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wrex 3060  {crab 3419  Vcvv 3462  cdif 3943  wss 3946  {csn 4623  {cpr 4625  cfv 6543  (class class class)co 7413  Vtxcvtx 28926  Edgcedg 28977   NeighbVtx cnbgr 29262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-nbgr 29263
This theorem is referenced by:  dfclnbgr4  47429  dfnbgr5  47451  dfnbgr6  47457
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