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Theorem nbgrssovtxOLD 26480
Description: Obsolete version of nbgrssovtx 26477 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
nbgrssovtxOLD (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))

Proof of Theorem nbgrssovtxOLD
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nbgrssovtxOLD.v . . . . 5 𝑉 = (Vtx‘𝐺)
21nbgrisvtxOLD 26457 . . . 4 ((𝐺𝑊𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣𝑉)
3 nbgrnself2OLD 26479 . . . . . . . . . 10 (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁))
43adantr 472 . . . . . . . . 9 ((𝐺𝑊𝑣 = 𝑁) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
5 df-nel 3036 . . . . . . . . . 10 (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
6 neleq1 3040 . . . . . . . . . . 11 (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
76adantl 473 . . . . . . . . . 10 ((𝐺𝑊𝑣 = 𝑁) → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
85, 7syl5bbr 274 . . . . . . . . 9 ((𝐺𝑊𝑣 = 𝑁) → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
94, 8mpbird 247 . . . . . . . 8 ((𝐺𝑊𝑣 = 𝑁) → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
109ex 449 . . . . . . 7 (𝐺𝑊 → (𝑣 = 𝑁 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
1110con2d 129 . . . . . 6 (𝐺𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → ¬ 𝑣 = 𝑁))
1211imp 444 . . . . 5 ((𝐺𝑊𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → ¬ 𝑣 = 𝑁)
1312neqned 2939 . . . 4 ((𝐺𝑊𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣𝑁)
14 eldifsn 4462 . . . 4 (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣𝑉𝑣𝑁))
152, 13, 14sylanbrc 701 . . 3 ((𝐺𝑊𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝑉 ∖ {𝑁}))
1615ex 449 . 2 (𝐺𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → 𝑣 ∈ (𝑉 ∖ {𝑁})))
1716ssrdv 3750 1 (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wnel 3035  cdif 3712  wss 3715  {csn 4321  cfv 6049  (class class class)co 6814  Vtxcvtx 26094   NeighbVtx cnbgr 26444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-nbgr 26445
This theorem is referenced by:  nbgrssvwo2OLD  26481
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