Theorem clnbgr0edg 47841

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 2730	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	dfclnbgr4 47829	. . 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 29291	. . . 4 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)
5	4	adantr 480	. . 3 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝐾) = ∅)
6	5	uneq2d 4134	. 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ (𝐺 NeighbVtx 𝐾)) = ({𝐾} ∪ ∅))
7		un0 4360	. . 3 ⊢ ({𝐾} ∪ ∅) = {𝐾}
8	7	a1i 11	. 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ ∅) = {𝐾})
9	3, 6, 8	3eqtrd 2769	1 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3915  ∅c0 4299  {csn 4592  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  Edgcedg 28981   NeighbVtx cnbgr 29266   ClNeighbVtx cclnbgr 47823

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-nbgr 29267  df-clnbgr 47824

This theorem is referenced by: (None)