Theorem nbgrssovtx 29306

Description: The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself. Stronger version of nbgrssvtx 29287. (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 29286	. . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ 𝑉)
3		nbgrnself2 29305	. . . . 5 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
4		df-nel 3036	. . . . . 6 ⊢ (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
5		neleq1 3041	. . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
6	4, 5	bitr3id 285	. . . . 5 ⊢ (𝑣 = 𝑋 → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
7	3, 6	mpbiri 258	. . . 4 ⊢ (𝑣 = 𝑋 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
8	7	necon2ai 2960	. . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ≠ 𝑋)
9		eldifsn 4766	. . 3 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑋))
10	2, 8, 9	sylanbrc 583	. 2 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ (𝑉 ∖ {𝑋}))
11	10	ssriv 3967	1 ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2107   ≠ wne 2931   ∉ wnel 3035   ∖ cdif 3928   ⊆ wss 3931  {csn 4606  ‘cfv 6541  (class class class)co 7413  Vtxcvtx 28941   NeighbVtx cnbgr 29277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-nbgr 29278

This theorem is referenced by:  nbgrssvwo2  29307  nbfusgrlevtxm1  29322  uvtxnbgr  29345  nbusgrvtxm1uvtx  29350  nbupgruvtxres  29352