Theorem nbgrssvwo2 29296

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 29295	. . 3 ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})
3		df-nel 3031	. . . . 5 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
4		disjsn 4678	. . . . 5 ⊢ (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
5	3, 4	sylbb2 238	. . . 4 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → ((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅)
6		reldisj 4419	. . . 4 ⊢ ((𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) → (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ (𝐺 NeighbVtx 𝑋) ⊆ ((𝑉 ∖ {𝑋}) ∖ {𝑀})))
7	5, 6	imbitrid 244	. . 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 4699	. . . 4 ⊢ {𝑀, 𝑋} = {𝑋, 𝑀}
10	9	difeq2i 4089	. . 3 ⊢ (𝑉 ∖ {𝑀, 𝑋}) = (𝑉 ∖ {𝑋, 𝑀})
11		difpr 4770	. . 3 ⊢ (𝑉 ∖ {𝑋, 𝑀}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
12	10, 11	eqtri 2753	. 2 ⊢ (𝑉 ∖ {𝑀, 𝑋}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
13	8, 12	sseqtrrdi 3991	1 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑀, 𝑋}))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∉ wnel 3030   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  {csn 4592  {cpr 4594  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930   NeighbVtx cnbgr 29266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-nbgr 29267

This theorem is referenced by:  nbfusgrlevtxm2  29312