Theorem nbgr0edg 28474

Description: In an empty graph (with no edges), every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)

Assertion

Ref	Expression
nbgr0edg	⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Proof of Theorem nbgr0edg

Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rzal 4499	. . . 4 ⊢ ((Edg‘𝐺) = ∅ → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝐾, 𝑛} ⊆ 𝑒)
2		ralnex 3071	. . . 4 ⊢ (∀𝑒 ∈ (Edg‘𝐺) ¬ {𝐾, 𝑛} ⊆ 𝑒 ↔ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
3	1, 2	sylib 217	. . 3 ⊢ ((Edg‘𝐺) = ∅ → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
4	3	ralrimivw 3149	. 2 ⊢ ((Edg‘𝐺) = ∅ → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
5	4	nbgr0vtxlem 28472	1 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541  ∀wral 3060  ∃wrex 3069   ∖ cdif 3938   ⊆ wss 3941  ∅c0 4315  {csn 4619  {cpr 4621  ‘cfv 6529  (class class class)co 7390  Vtxcvtx 28116  Edgcedg 28167   NeighbVtx cnbgr 28449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7705

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6481  df-fun 6531  df-fv 6537  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7954  df-2nd 7955  df-nbgr 28450

This theorem is referenced by:  uvtx01vtx  28514