Theorem nbgr0vtx 29188

Description: In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.) (Proof shortened by AV, 10-May-2025.)

Assertion

Ref	Expression
nbgr0vtx	⊢ ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Proof of Theorem nbgr0vtx

Step	Hyp	Ref	Expression
1		nel02 4336	. . 3 ⊢ ((Vtx‘𝐺) = ∅ → ¬ 𝐾 ∈ (Vtx‘𝐺))
2		df-nel 3044	. . 3 ⊢ (𝐾 ∉ (Vtx‘𝐺) ↔ ¬ 𝐾 ∈ (Vtx‘𝐺))
3	1, 2	sylibr 233	. 2 ⊢ ((Vtx‘𝐺) = ∅ → 𝐾 ∉ (Vtx‘𝐺))
4		eqid 2728	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	nbgrnvtx0 29172	. 2 ⊢ (𝐾 ∉ (Vtx‘𝐺) → (𝐺 NeighbVtx 𝐾) = ∅)
6	3, 5	syl 17	1 ⊢ ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2098   ∉ wnel 3043  ∅c0 4326  ‘cfv 6553  (class class class)co 7426  Vtxcvtx 28829   NeighbVtx cnbgr 29165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-nbgr 29166

This theorem is referenced by: (None)