Theorem nbgrssovtx 40581

Description: The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 40577. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.)

Hypothesis

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

Assertion

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

Proof of Theorem nbgrssovtx

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgrssovtx.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrisvtx 40576	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ 𝑉)
3		nbgrnself2 40580	. . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑊 → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
4	3	adantr 479	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))
5		df-nel 2782	. . . . . . . . . 10 ⊢ (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
6		neleq1 2887	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
7	6	adantl 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → (𝑣 ∉ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
8	5, 7	syl5bbr 272	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → (¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
9	4, 8	mpbird 245	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 = 𝑁) → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
10	9	ex 448	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → (𝑣 = 𝑁 → ¬ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
11	10	con2d 127	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → ¬ 𝑣 = 𝑁))
12	11	imp 443	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → ¬ 𝑣 = 𝑁)
13	12	neqned 2788	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ≠ 𝑁)
14		eldifsn 4259	. . . 4 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣 ∈ 𝑉 ∧ 𝑣 ≠ 𝑁))
15	2, 13, 14	sylanbrc 694	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝑉 ∖ {𝑁}))
16	15	ex 448	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) → 𝑣 ∈ (𝑉 ∖ {𝑁})))
17	16	ssrdv 3573	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 194   ∧ wa 382   = wceq 1474   ∈ wcel 1976   ≠ wne 2779   ∉ wnel 2780   ∖ cdif 3536   ⊆ wss 3539  {csn 4124  ‘cfv 5790  (class class class)co 6527  Vtxcvtx 40224   NeighbVtx cnbgr 40545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-nbgr 40549

This theorem is referenced by:  nbgrssvwo2  40582  usgrnbssovtx  40584  nbfusgrlevtxm1  40600  uvtxnbgr  40622  nbusgrvtxm1uvtx  40627  nbupgruvtxres  40629