Theorem nbuhgr 26694

Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 26687 (with 𝑁 being an arbitrary set instead of being a vertex) only holds for classes whose edges are subsets of the set of vertices (hypergraphs!). (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)

Hypotheses

Ref	Expression
nbuhgr.v	⊢ 𝑉 = (Vtx‘𝐺)
nbuhgr.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
nbuhgr	⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺,𝑛   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑒,𝑋,𝑛

Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem nbuhgr

Step	Hyp	Ref	Expression
1		nbuhgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		nbuhgr.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	nbgrval 26687	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
4	3	a1d 25	. 2 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
5		df-nel 3076	. . . . . 6 ⊢ (𝑁 ∉ 𝑉 ↔ ¬ 𝑁 ∈ 𝑉)
6	1	nbgrnvtx0 26690	. . . . . 6 ⊢ (𝑁 ∉ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
7	5, 6	sylbir 227	. . . . 5 ⊢ (¬ 𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
8	7	adantr 474	. . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝐺 NeighbVtx 𝑁) = ∅)
9		simpl 476	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → 𝐺 ∈ UHGraph)
10	9	adantr 474	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝐺 ∈ UHGraph)
11	2	eleq2i 2851	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
12	11	biimpi 208	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
13		edguhgr 26481	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
14	10, 12, 13	syl2an 589	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
15		selpw 4386	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ (Vtx‘𝐺))
16	1	eqcomi 2787	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = 𝑉
17	16	sseq2i 3849	. . . . . . . . . . . 12 ⊢ (𝑒 ⊆ (Vtx‘𝐺) ↔ 𝑒 ⊆ 𝑉)
18	15, 17	bitri 267	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ 𝑉)
19		sstr 3829	. . . . . . . . . . . . . . 15 ⊢ (({𝑁, 𝑛} ⊆ 𝑒 ∧ 𝑒 ⊆ 𝑉) → {𝑁, 𝑛} ⊆ 𝑉)
20		prssg 4583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝑛 ∈ V) → ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ↔ {𝑁, 𝑛} ⊆ 𝑉))
21	20	bicomd 215	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝑛 ∈ V) → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
22	21	elvd 3403	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ 𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
23		simpl 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → 𝑁 ∈ 𝑉)
24	22, 23	syl6bi 245	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ 𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 → 𝑁 ∈ 𝑉))
25	19, 24	syl5com 31	. . . . . . . . . . . . . 14 ⊢ (({𝑁, 𝑛} ⊆ 𝑒 ∧ 𝑒 ⊆ 𝑉) → (𝑁 ∈ 𝑋 → 𝑁 ∈ 𝑉))
26	25	ex 403	. . . . . . . . . . . . 13 ⊢ ({𝑁, 𝑛} ⊆ 𝑒 → (𝑒 ⊆ 𝑉 → (𝑁 ∈ 𝑋 → 𝑁 ∈ 𝑉)))
27	26	com13 88	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ 𝑋 → (𝑒 ⊆ 𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
28	27	ad3antlr 721	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → (𝑒 ⊆ 𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
29	18, 28	syl5bi 234	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
30	14, 29	mpd 15	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉))
31	30	rexlimdva 3213	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉))
32	31	con3rr3 153	. . . . . . 7 ⊢ (¬ 𝑁 ∈ 𝑉 → (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
33	32	expdimp 446	. . . . . 6 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝑛 ∈ (𝑉 ∖ {𝑁}) → ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
34	33	ralrimiv 3147	. . . . 5 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒)
35		rabeq0 4187	. . . . 5 ⊢ ({𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒)
36	34, 35	sylibr 226	. . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅)
37	8, 36	eqtr4d 2817	. . 3 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
38	37	ex 403	. 2 ⊢ (¬ 𝑁 ∈ 𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
39	4, 38	pm2.61i 177	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ∉ wnel 3075  ∀wral 3090  ∃wrex 3091  {crab 3094  Vcvv 3398   ∖ cdif 3789   ⊆ wss 3792  ∅c0 4141  𝒫 cpw 4379  {csn 4398  {cpr 4400  ‘cfv 6137  (class class class)co 6924  Vtxcvtx 26348  Edgcedg 26399  UHGraphcuhgr 26408   NeighbVtx cnbgr 26683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-edg 26400  df-uhgr 26410  df-nbgr 26684

This theorem is referenced by:  uhgrnbgr0nb  26705