Theorem nbgrnvtx0 29540

Description: If a class 𝑋 is not a vertex of a graph 𝐺, then it has no neighbors in 𝐺. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)

Hypothesis

Ref	Expression
nbgrel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
nbgrnvtx0	⊢ (𝑋 ∉ 𝑉 → (𝐺 NeighbVtx 𝑋) = ∅)

Proof of Theorem nbgrnvtx0

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

Step	Hyp	Ref	Expression
1		nbgrel.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2		csbfv 6914	. . . . . 6 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺)
3	1, 2	eqtr4i 2788	. . . . 5 ⊢ 𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)
4		neleq2 3068	. . . . 5 ⊢ (𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → (𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
5	3, 4	ax-mp 5	. . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))
6	5	biimpi 218	. . 3 ⊢ (𝑋 ∉ 𝑉 → 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))
7	6	olcd 885	. 2 ⊢ (𝑋 ∉ 𝑉 → (𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)))
8		df-nbgr 29534	. . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
9	8	mpoxneldm 8192	. 2 ⊢ ((𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) → (𝐺 NeighbVtx 𝑋) = ∅)
10	7, 9	syl 17	1 ⊢ (𝑋 ∉ 𝑉 → (𝐺 NeighbVtx 𝑋) = ∅)

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 858   = wceq 1560   ∉ wnel 3061  ∃wrex 3086  {crab 3414  Vcvv 3454  ⦋csb 3852   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  {csn 4582  {cpr 4584  ‘cfv 6521  (class class class)co 7396  Vtxcvtx 29197  Edgcedg 29248   NeighbVtx cnbgr 29533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-nbgr 29534

This theorem is referenced by:  nbuhgr  29544  nbumgr  29548  nbgr0vtx  29556  nbgr0edglem  29557  nbgr1vtx  29559