Theorem clnbgr0edg 48754

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 2760	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	dfclnbgr4 48741	. . 3 ⊢ (𝐾 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝐾) = ({𝐾} ∪ (𝐺 NeighbVtx 𝐾)))
3	2	adantl 487	. 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = ({𝐾} ∪ (𝐺 NeighbVtx 𝐾)))
4		nbgr0edg 29818	. . . 4 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)
5	4	adantr 486	. . 3 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝐾) = ∅)
6	5	uneq2d 4115	. 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ (𝐺 NeighbVtx 𝐾)) = ({𝐾} ∪ ∅))
7		un0 4344	. . 3 ⊢ ({𝐾} ∪ ∅) = {𝐾}
8	7	a1i 11	. 2 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → ({𝐾} ∪ ∅) = {𝐾})
9	3, 6, 8	3eqtrd 2799	1 ⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∪ cun 3897  ∅c0 4279  {csn 4584  ‘cfv 6533  (class class class)co 7414  Vtxcvtx 29454  Edgcedg 29505   NeighbVtx cnbgr 29793   ClNeighbVtx cclnbgr 48735

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-nbgr 29794  df-clnbgr 48736

This theorem is used by: (None)