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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 ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))

Proof of Theorem nbgrssvwo2OLD
StepHypRef Expression
1 nbgrssovtxOLD.v . . . . 5 𝑉 = (Vtx‘𝐺)
21nbgrssovtxOLD 26451 . . . 4 (𝐺𝑊 → (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))
53, 4sylbb2 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})))
75, 6syl5ib 234 . . . 4 ((𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})))
82, 7syl 17 . . 3 (𝐺𝑊 → (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})))
98imp 444 . 2 ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))
10 prcom 4403 . . . 4 {𝑀, 𝑁} = {𝑁, 𝑀}
1110difeq2i 3860 . . 3 (𝑉 ∖ {𝑀, 𝑁}) = (𝑉 ∖ {𝑁, 𝑀})
12 difpr 4471 . . 3 (𝑉 ∖ {𝑁, 𝑀}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀})
1311, 12eqtri 2774 . 2 (𝑉 ∖ {𝑀, 𝑁}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀})
149, 13syl6sseqr 3785 1 ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))
Colors of variables: wff setvar class
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)
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