Theorem clnbgrsym 48336

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 461	. . 3 ⊢ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ↔ (𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)))
2		eqcom 2747	. . . 4 ⊢ (𝑁 = 𝐾 ↔ 𝐾 = 𝑁)
3		prcom 4671	. . . . . 6 ⊢ {𝐾, 𝑁} = {𝑁, 𝐾}
4	3	sseq1i 3950	. . . . 5 ⊢ ({𝐾, 𝑁} ⊆ 𝑒 ↔ {𝑁, 𝐾} ⊆ 𝑒)
5	4	rexbii 3087	. . . 4 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒)
6	2, 5	orbi12i 920	. . 3 ⊢ ((𝑁 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) ↔ (𝐾 = 𝑁 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒))
7	1, 6	anbi12i 634	. 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 2740	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2740	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
10	8, 9	clnbgrel 48326	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ (𝑁 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒)))
11	8, 9	clnbgrel 48326	. 2 ⊢ (𝐾 ∈ (𝐺 ClNeighbVtx 𝑁) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ (𝐾 = 𝑁 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒)))
12	7, 10, 11	3bitr4i 304	1 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 ClNeighbVtx 𝑁))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∃wrex 3064   ⊆ wss 3890  {cpr 4564  ‘cfv 6492  (class class class)co 7363  Vtxcvtx 29090  Edgcedg 29141   ClNeighbVtx cclnbgr 48316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-clnbgr 48317

This theorem is referenced by: (None)