Theorem clnbgrsym 48192

Description: In a graph, the closed 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 AV, 10-May-2025.)

Assertion

Ref	Expression
clnbgrsym	⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 ClNeighbVtx 𝑁))

Proof of Theorem clnbgrsym

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		eqcom 2744	. . . 4 ⊢ (𝑁 = 𝐾 ↔ 𝐾 = 𝑁)
3		prcom 4691	. . . . . 6 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾}
4	3	sseq1i 3964	. . . . 5 ⊢ ({𝐾, 𝑁} ⊆ 𝑒 ↔ {𝑁, 𝐾} ⊆ 𝑒)
5	4	rexbii 3085	. . . 4 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒)
6	2, 5	orbi12i 915	. . 3 ⊢ ((𝑁 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) ↔ (𝐾 = 𝑁 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒))
7	1, 6	anbi12i 629	. 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)))
8		eqid 2737	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2737	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
10	8, 9	clnbgrel 48182	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ (𝑁 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒)))
11	8, 9	clnbgrel 48182	. 2 ⊢ (𝐾 ∈ (𝐺 ClNeighbVtx 𝑁) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ (𝐾 = 𝑁 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒)))
12	7, 10, 11	3bitr4i 303	1 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 ClNeighbVtx 𝑁))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903  {cpr 4584  ‘cfv 6500  (class class class)co 7368  Vtxcvtx 29081  Edgcedg 29132   ClNeighbVtx cclnbgr 48172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-clnbgr 48173

This theorem is referenced by: (None)