Theorem nbgr0vtx 26252

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

Assertion

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

Proof of Theorem nbgr0vtx

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

Step	Hyp	Ref	Expression
1		ral0 4076	. . 3 ⊢ ∀𝑛 ∈ ∅ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒
2		difeq1 3721	. . . . 5 ⊢ ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = (∅ ∖ {𝐾}))
3		0dif 3977	. . . . 5 ⊢ (∅ ∖ {𝐾}) = ∅
4	2, 3	syl6eq 2672	. . . 4 ⊢ ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = ∅)
5	4	raleqdv 3144	. . 3 ⊢ ((Vtx‘𝐺) = ∅ → (∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒 ↔ ∀𝑛 ∈ ∅ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒))
6	1, 5	mpbiri 248	. 2 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
7	6	nbgr0vtxlem 26251	1 ⊢ ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1483  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ‘cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   NeighbVtx cnbgr 26224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-nbgr 26228

This theorem is referenced by: (None)