Theorem nbgr0edg 29938

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 4450	. . . 4 ⊢ ((Edg‘𝐺) = ∅ → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝐾, 𝑛} ⊆ 𝑒)
2		ralnex 3089	. . . 4 ⊢ (∀𝑒 ∈ (Edg‘𝐺) ¬ {𝐾, 𝑛} ⊆ 𝑒 ↔ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
3	1, 2	sylib 221	. . 3 ⊢ ((Edg‘𝐺) = ∅ → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
4	3	ralrimivw 3159	. 2 ⊢ ((Edg‘𝐺) = ∅ → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
5	4	nbgr0edglem 29937	1 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∀wral 3077  ∃wrex 3087   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  {csn 4584  {cpr 4586  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29574  Edgcedg 29625   NeighbVtx cnbgr 29913

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 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-nbgr 29914

This theorem is used by:  uvtx01vtx  29978  clnbgr0edg  48934