Theorem nbgrsym 29169

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 2990	. . 3 ⊢ (𝑁 ≠ 𝐾 ↔ 𝐾 ≠ 𝑁)
3		prcom 4732	. . . . 5 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾}
4	3	sseq1i 4006	. . . 4 ⊢ ({𝐾, 𝑁} ⊆ 𝑒 ↔ {𝑁, 𝐾} ⊆ 𝑒)
5	4	rexbii 3090	. . 3 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒)
6	1, 2, 5	3anbi123i 1153	. 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 2728	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8		eqid 2728	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
9	7, 8	nbgrel 29146	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒))
10	7, 8	nbgrel 29146	. 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 205   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2099   ≠ wne 2936  ∃wrex 3066   ⊆ wss 3945  {cpr 4626  ‘cfv 6542  (class class class)co 7414  Vtxcvtx 28802  Edgcedg 28853   NeighbVtx cnbgr 29138

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 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-nbgr 29139

This theorem is referenced by:  nbusgredgeu0  29174  uvtxnbgrvtx  29199  cplgr3v  29241  frgrncvvdeqlem1  30102  frgrwopreglem4a  30113  numclwwlk1lem2foa  30157

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