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	⊢ (𝐺 ∈ 𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))

Proof of Theorem nbgrssovtxOLD

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgrssovtxOLD.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrisvtxOLD 26457	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ 𝑉)
3		nbgrnself2OLD 26479	. . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
4	3	adantr 472	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 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)))
7	6	adantl 473	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
8	5, 7	syl5bbr 274	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
9	4, 8	mpbird 247	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
10	9	ex 449	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (𝑣 = 𝑁 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
11	10	con2d 129	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → ¬ 𝑣 = 𝑁))
12	11	imp 444	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → ¬ 𝑣 = 𝑁)
13	12	neqned 2939	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ≠ 𝑁)
14		eldifsn 4462	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁))
15	2, 13, 14	sylanbrc 701	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝑉 ∖ {𝑁}))
16	15	ex 449	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → 𝑣 ∈ (𝑉 ∖ {𝑁})))
17	16	ssrdv 3750	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))

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