Theorem nbgrssvwo2 29943

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 29942	. . 3 ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})
3		df-nel 3062	. . . . 5 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
4		disjsn 4671	. . . . 5 ⊢ (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑋))
5	3, 4	sylbb2 241	. . . 4 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → ((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅)
6		reldisj 4405	. . . 4 ⊢ ((𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) → (((𝐺 NeighbVtx 𝑋) ∩ {𝑀}) = ∅ ↔ (𝐺 NeighbVtx 𝑋) ⊆ ((𝑉 ∖ {𝑋}) ∖ {𝑀})))
7	5, 6	imbitrid 247	. . 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 4692	. . . 4 ⊢ {𝑀, 𝑋} = {𝑋, 𝑀}
10	9	difeq2i 4060	. . 3 ⊢ (𝑉 ∖ {𝑀, 𝑋}) = (𝑉 ∖ {𝑋, 𝑀})
11		difpr 4765	. . 3 ⊢ (𝑉 ∖ {𝑋, 𝑀}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
12	10, 11	eqtri 2783	. 2 ⊢ (𝑉 ∖ {𝑀, 𝑋}) = ((𝑉 ∖ {𝑋}) ∖ {𝑀})
13	8, 12	sseqtrrdi 3949	1 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑀, 𝑋}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144   ∉ wnel 3061   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  {csn 4583  {cpr 4585  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29574   NeighbVtx cnbgr 29913

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-nbgr 29914

This theorem is used by:  nbfusgrlevtxm2  29959