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Theorem nbgrasym 25968
 Description: A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgrasym (𝑉 USGrph 𝐸 → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ 𝐾 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))

Proof of Theorem nbgrasym
StepHypRef Expression
1 usgrav 25867 . 2 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2 3ancoma 1038 . . . . 5 ((𝐾𝑉𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝑁𝑉𝐾𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸))
3 prcom 4211 . . . . . . 7 {𝐾, 𝑁} = {𝑁, 𝐾}
43eleq1i 2679 . . . . . 6 ({𝐾, 𝑁} ∈ ran 𝐸 ↔ {𝑁, 𝐾} ∈ ran 𝐸)
543anbi3i 1248 . . . . 5 ((𝑁𝑉𝐾𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝑁𝑉𝐾𝑉 ∧ {𝑁, 𝐾} ∈ ran 𝐸))
62, 5bitri 263 . . . 4 ((𝐾𝑉𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝑁𝑉𝐾𝑉 ∧ {𝑁, 𝐾} ∈ ran 𝐸))
76a1i 11 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝐾𝑉𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝑁𝑉𝐾𝑉 ∧ {𝑁, 𝐾} ∈ ran 𝐸)))
8 nbgrael 25955 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝐾𝑉𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))
9 nbgrael 25955 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐾 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ (𝑁𝑉𝐾𝑉 ∧ {𝑁, 𝐾} ∈ ran 𝐸)))
107, 8, 93bitr4d 299 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ 𝐾 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
111, 10syl 17 1 (𝑉 USGrph 𝐸 → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ 𝐾 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  Vcvv 3173  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-usgra 25862  df-nbgra 25949 This theorem is referenced by:  uvtxnbgravtx  26023
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