Theorem nbgrsym 29290

Description: In a graph, the neighborhood relation is symmetric: a vertex 𝑁 in a graph 𝐺 is a neighbor of a second vertex 𝐾 iff the second vertex 𝐾 is a neighbor of the first vertex 𝑁. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Revised by AV, 12-Feb-2022.)

Assertion

Ref	Expression
nbgrsym	⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁))

Proof of Theorem nbgrsym

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ancom 460	. . 3 ⊢ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ↔ (𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)))
2		necom 2978	. . 3 ⊢ (𝑁 ≠ 𝐾 ↔ 𝐾 ≠ 𝑁)
3		prcom 4696	. . . . 5 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾}
4	3	sseq1i 3975	. . . 4 ⊢ ({𝐾, 𝑁} ⊆ 𝑒 ↔ {𝑁, 𝐾} ⊆ 𝑒)
5	4	rexbii 3076	. . 3 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒)
6	1, 2, 5	3anbi123i 1155	. 2 ⊢ (((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ 𝐾 ≠ 𝑁 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒))
7		eqid 2729	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8		eqid 2729	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
9	7, 8	nbgrel 29267	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒))
10	7, 8	nbgrel 29267	. 2 ⊢ (𝐾 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ 𝐾 ≠ 𝑁 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒))
11	6, 9, 10	3bitr4i 303	1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ⊆ wss 3914  {cpr 4591  ‘cfv 6511  (class class class)co 7387  Vtxcvtx 28923  Edgcedg 28974   NeighbVtx cnbgr 29259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-nbgr 29260

This theorem is referenced by:  nbusgredgeu0  29295  uvtxnbgrvtx  29320  cplgr3v  29362  frgrncvvdeqlem1  30228  frgrwopreglem4a  30239  numclwwlk1lem2foa  30283