Theorem nbgrssovtx 29722

Description: The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself. Stronger version of nbgrssvtx 29703. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.)

Hypothesis

Ref	Expression
nbgrssovtx.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
nbgrssovtx	⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})

Proof of Theorem nbgrssovtx

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgrssovtx.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrisvtx 29702	. . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ 𝑉)
3		nbgrnself2 29721	. . . . 5 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
4		df-nel 3064	. . . . . 6 ⊢ (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
5		neleq1 3069	. . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
6	4, 5	bitr3id 288	. . . . 5 ⊢ (𝑣 = 𝑋 → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
7	3, 6	mpbiri 261	. . . 4 ⊢ (𝑣 = 𝑋 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
8	7	necon2ai 2986	. . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ≠ 𝑋)
9		eldifsn 4752	. . 3 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑋))
10	2, 8, 9	sylanbrc 594	. 2 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ (𝑉 ∖ {𝑋}))
11	10	ssriv 3940	1 ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2142   ≠ wne 2957   ∉ wnel 3063   ∖ cdif 3901   ⊆ wss 3904  {csn 4588  ‘cfv 6536  (class class class)co 7412  Vtxcvtx 29357   NeighbVtx cnbgr 29693

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-nbgr 29694

This theorem is used by:  nbgrssvwo2  29723  nbfusgrlevtxm1  29738  uvtxnbgr  29761  nbusgrvtxm1uvtx  29766  nbupgruvtxres  29768