Theorem nbgrssovtx 27631

Description: The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself. Stronger version of nbgrssvtx 27612. (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 27611	. . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ 𝑉)
3		nbgrnself2 27630	. . . . 5 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
4		df-nel 3049	. . . . . 6 ⊢ (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
5		neleq1 3053	. . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
6	4, 5	bitr3id 284	. . . . 5 ⊢ (𝑣 = 𝑋 → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
7	3, 6	mpbiri 257	. . . 4 ⊢ (𝑣 = 𝑋 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑋))
8	7	necon2ai 2972	. . 3 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ≠ 𝑋)
9		eldifsn 4717	. . 3 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑋))
10	2, 8, 9	sylanbrc 582	. 2 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑋) → 𝑣 ∈ (𝑉 ∖ {𝑋}))
11	10	ssriv 3921	1 ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ∉ wnel 3048   ∖ cdif 3880   ⊆ wss 3883  {csn 4558  ‘cfv 6418  (class class class)co 7255  Vtxcvtx 27269   NeighbVtx cnbgr 27602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-nbgr 27603

This theorem is referenced by:  nbgrssvwo2  27632  nbfusgrlevtxm1  27647  uvtxnbgr  27670  nbusgrvtxm1uvtx  27675  nbupgruvtxres  27677