Theorem nbgrssvwo2 27729

Description: The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself and a class 𝑀 which is not a neighbor of 𝑋. (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
nbgrssvwo2	⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑀, 𝑋}))

Proof of Theorem nbgrssvwo2

Step	Hyp	Ref	Expression
1		nbgrssovtx.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrssovtx 27728	. . 3 ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})
3		df-nel 3050	. . . . 5 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
4		disjsn 4647	. . . . 5 ⊢ (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
5	3, 4	sylbb2 237	. . . 4 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → ((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅)
6		reldisj 4385	. . . 4 ⊢ ((𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) → (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ (𝐺 NeighbVtx 𝑋) ⊆ ((𝑉 ∖ {𝑋}) ∖ {𝑀})))
7	5, 6	syl5ib 243	. . 3 ⊢ ((𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) → (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ ((𝑉 ∖ {𝑋}) ∖ {𝑀})))
8	2, 7	ax-mp 5	. 2 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ ((𝑉 ∖ {𝑋}) ∖ {𝑀}))
9		prcom 4668	. . . 4 ⊢ {𝑀, 𝑋} = {𝑋, 𝑀}
10	9	difeq2i 4054	. . 3 ⊢ (𝑉 ∖ {𝑀, 𝑋}) = (𝑉 ∖ {𝑋, 𝑀})
11		difpr 4736	. . 3 ⊢ (𝑉 ∖ {𝑋, 𝑀}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
12	10, 11	eqtri 2766	. 2 ⊢ (𝑉 ∖ {𝑀, 𝑋}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
13	8, 12	sseqtrrdi 3972	1 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑀, 𝑋}))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2106   ∉ wnel 3049   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561  {cpr 4563  ‘cfv 6433  (class class class)co 7275  Vtxcvtx 27366   NeighbVtx cnbgr 27699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-nbgr 27700

This theorem is referenced by:  nbfusgrlevtxm2  27745