Theorem nbgr0edg 29336

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 4484	. . . 4 ⊢ ((Edg‘𝐺) = ∅ → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝐾, 𝑛} ⊆ 𝑒)
2		ralnex 3062	. . . 4 ⊢ (∀𝑒 ∈ (Edg‘𝐺) ¬ {𝐾, 𝑛} ⊆ 𝑒 ↔ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
3	1, 2	sylib 218	. . 3 ⊢ ((Edg‘𝐺) = ∅ → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
4	3	ralrimivw 3136	. 2 ⊢ ((Edg‘𝐺) = ∅ → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
5	4	nbgr0edglem 29335	1 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540  ∀wral 3051  ∃wrex 3060   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  {csn 4601  {cpr 4603  ‘cfv 6531  (class class class)co 7405  Vtxcvtx 28975  Edgcedg 29026   NeighbVtx cnbgr 29311

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-nbgr 29312

This theorem is referenced by:  uvtx01vtx  29376  clnbgr0edg  47850