Theorem nbuhgr 29416

Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 29409 (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 29409	. . 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 3037	. . . . . 6 ⊢ (𝑁 ∉ 𝑉 ↔ ¬ 𝑁 ∈ 𝑉)
6	1	nbgrnvtx0 29412	. . . . . 6 ⊢ (𝑁 ∉ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
7	5, 6	sylbir 235	. . . . 5 ⊢ (¬ 𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
8	7	adantr 480	. . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝐺 NeighbVtx 𝑁) = ∅)
9		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → 𝐺 ∈ UHGraph)
10	9	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝐺 ∈ UHGraph)
11	2	eleq2i 2828	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
12	11	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
13		edguhgr 29202	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
14	10, 12, 13	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
15		velpw 4559	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ (Vtx‘𝐺))
16	1	eqcomi 2745	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = 𝑉
17	16	sseq2i 3963	. . . . . . . . . . . 12 ⊢ (𝑒 ⊆ (Vtx‘𝐺) ↔ 𝑒 ⊆ 𝑉)
18	15, 17	bitri 275	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ 𝑉)
19		sstr 3942	. . . . . . . . . . . . . . 15 ⊢ (({𝑁, 𝑛} ⊆ 𝑒 ∧ 𝑒 ⊆ 𝑉) → {𝑁, 𝑛} ⊆ 𝑉)
20		prssg 4775	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝑛 ∈ V) → ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ↔ {𝑁, 𝑛} ⊆ 𝑉))
21	20	bicomd 223	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝑛 ∈ V) → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
22	21	elvd 3446	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ 𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
23		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → 𝑁 ∈ 𝑉)
24	22, 23	biimtrdi 253	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 731	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → (𝑒 ⊆ 𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
29	18, 28	biimtrid 242	. . . . . . . . . 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 3137	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉))
32	31	con3rr3 155	. . . . . . 7 ⊢ (¬ 𝑁 ∈ 𝑉 → (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
33	32	expdimp 452	. . . . . 6 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝑛 ∈ (𝑉 ∖ {𝑁}) → ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
34	33	ralrimiv 3127	. . . . 5 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒)
35		rabeq0 4340	. . . . 5 ⊢ ({𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒)
36	34, 35	sylibr 234	. . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅)
37	8, 36	eqtr4d 2774	. . 3 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
38	37	ex 412	. 2 ⊢ (¬ 𝑁 ∈ 𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
39	4, 38	pm2.61i 182	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∉ wnel 3036  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  {cpr 4582  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29069  Edgcedg 29120  UHGraphcuhgr 29129   NeighbVtx cnbgr 29405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-edg 29121  df-uhgr 29131  df-nbgr 29406

This theorem is referenced by:  uhgrnbgr0nb  29427