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Theorem nbgrssvwo2OLD 26603
Description: Obsolete version of nbgrssvwo2 26600 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 26602 . . . 4 (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))
3 df-nel 3075 . . . . . 6 (𝑀 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑁))
4 disjsn 4436 . . . . . 6 (((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑁))
53, 4sylbb2 230 . . . . 5 (𝑀 ∉ (𝐺 NeighbVtx 𝑁) → ((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅)
6 reldisj 4215 . . . . 5 ((𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (((𝐺 NeighbVtx 𝑁) ∩ {𝑀}) = ∅ ↔ (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})))
75, 6syl5ib 236 . . . 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 396 . 2 ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))
10 prcom 4456 . . . 4 {𝑀, 𝑁} = {𝑁, 𝑀}
1110difeq2i 3923 . . 3 (𝑉 ∖ {𝑀, 𝑁}) = (𝑉 ∖ {𝑁, 𝑀})
12 difpr 4522 . . 3 (𝑉 ∖ {𝑁, 𝑀}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀})
1311, 12eqtri 2821 . 2 (𝑉 ∖ {𝑀, 𝑁}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀})
149, 13syl6sseqr 3848 1 ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wcel 2157  wnel 3074  cdif 3766  cin 3768  wss 3769  c0 4115  {csn 4368  {cpr 4370  cfv 6101  (class class class)co 6878  Vtxcvtx 26231   NeighbVtx cnbgr 26566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-nbgr 26567
This theorem is referenced by: (None)
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