Theorem nbgrssvwo2 26303

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 26302	. . 3 ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})
3		df-nel 2927	. . . . 5 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
4		disjsn 4278	. . . . 5 ⊢ (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
5	3, 4	sylbb2 228	. . . 4 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → ((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅)
6		reldisj 4053	. . . 4 ⊢ ((𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) → (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ (𝐺 NeighbVtx 𝑋) ⊆ ((𝑉 ∖ {𝑋}) ∖ {𝑀})))
7	5, 6	syl5ib 234	. . 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 4299	. . . 4 ⊢ {𝑀, 𝑋} = {𝑋, 𝑀}
10	9	difeq2i 3758	. . 3 ⊢ (𝑉 ∖ {𝑀, 𝑋}) = (𝑉 ∖ {𝑋, 𝑀})
11		difpr 4366	. . 3 ⊢ (𝑉 ∖ {𝑋, 𝑀}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
12	10, 11	eqtri 2673	. 2 ⊢ (𝑉 ∖ {𝑀, 𝑋}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
13	8, 12	syl6sseqr 3685	1 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑀, 𝑋}))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ∈ wcel 2030   ∉ wnel 2926   ∖ cdif 3604   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {csn 4210  {cpr 4212  ‘cfv 5926  (class class class)co 6690  Vtxcvtx 25919   NeighbVtx cnbgr 26269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-nbgr 26270

This theorem is referenced by:  nbfusgrlevtxm2  26324