Theorem nbgr0vtx 26475

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 4218	. . 3 ⊢ ∀𝑛 ∈ ∅ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒
2		difeq1 3872	. . . . 5 ⊢ ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = (∅ ∖ {𝐾}))
3		0dif 4122	. . . . 5 ⊢ (∅ ∖ {𝐾}) = ∅
4	2, 3	syl6eq 2821	. . . 4 ⊢ ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = ∅)
5	4	raleqdv 3293	. . 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 26474	1 ⊢ ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1631  ∀wral 3061  ∃wrex 3062   ∖ cdif 3720   ⊆ wss 3723  ∅c0 4063  {csn 4317  {cpr 4319  ‘cfv 6030  (class class class)co 6796  Vtxcvtx 26095  Edgcedg 26160   NeighbVtx cnbgr 26447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-nbgr 26448

This theorem is referenced by: (None)