Theorem clnbgr0edg 47709

Description: In an empty graph (with no edges), all closed neighborhoods consists of a single vertex. (Contributed by AV, 10-May-2025.)

Assertion

Ref	Expression
clnbgr0edg	⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾})

Proof of Theorem clnbgr0edg

Step	Hyp	Ref	Expression
1		eqid 2740	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	dfclnbgr4 47698	. . 3 ⊢ (𝐾 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝐾) = ({𝐾} ∪ (𝐺 NeighbVtx 𝐾)))
3	2	adantl 481	. 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = ({𝐾} ∪ (𝐺 NeighbVtx 𝐾)))
4		nbgr0edg 29392	. . . 4 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)
5	4	adantr 480	. . 3 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝐾) = ∅)
6	5	uneq2d 4191	. 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ (𝐺 NeighbVtx 𝐾)) = ({𝐾} ∪ ∅))
7		un0 4417	. . 3 ⊢ ({𝐾} ∪ ∅) = {𝐾}
8	7	a1i 11	. 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ ∅) = {𝐾})
9	3, 6, 8	3eqtrd 2784	1 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∪ cun 3974  ∅c0 4352  {csn 4648  ‘cfv 6573  (class class class)co 7448  Vtxcvtx 29031  Edgcedg 29082   NeighbVtx cnbgr 29367   ClNeighbVtx cclnbgr 47692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-nbgr 29368  df-clnbgr 47693

This theorem is referenced by: (None)