Theorem nbgrssvwo2OLD 26452

Description: Obsolete version of nbgrssvwo2 26449 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
nbgrssovtxOLD.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
nbgrssvwo2OLD	⊢ ((𝐺 ∈ 𝑊 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))

Proof of Theorem nbgrssvwo2OLD

Step	Hyp	Ref	Expression
1		nbgrssovtxOLD.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrssovtxOLD 26451	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))
3		df-nel 3028	. . . . . 6 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑁))
4		disjsn 4382	. . . . . 6 ⊢ (((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑁))
5	3, 4	sylbb2 228	. . . . 5 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → ((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅)
6		reldisj 4155	. . . . 5 ⊢ ((𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅ ↔ (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})))
7	5, 6	syl5ib 234	. . . 4 ⊢ ((𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})))
8	2, 7	syl 17	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})))
9	8	imp 444	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))
10		prcom 4403	. . . 4 ⊢ {𝑀, 𝑁} = {𝑁, 𝑀}
11	10	difeq2i 3860	. . 3 ⊢ (𝑉 ∖ {𝑀, 𝑁}) = (𝑉 ∖ {𝑁, 𝑀})
12		difpr 4471	. . 3 ⊢ (𝑉 ∖ {𝑁, 𝑀}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀})
13	11, 12	eqtri 2774	. 2 ⊢ (𝑉 ∖ {𝑀, 𝑁}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀})
14	9, 13	syl6sseqr 3785	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1624   ∈ wcel 2131   ∉ wnel 3027   ∖ cdif 3704   ∩ cin 3706   ⊆ wss 3707  ∅c0 4050  {csn 4313  {cpr 4315  ‘cfv 6041  (class class class)co 6805  Vtxcvtx 26065   NeighbVtx cnbgr 26415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-nbgr 26416

This theorem is referenced by: (None)