Theorem uhgrnbgr0nb 29992

Description: A vertex which is not endpoint of an edge has no neighbor in a hypergraph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)

Assertion

Ref	Expression
uhgrnbgr0nb	⊢ ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → (𝐺 NeighbVtx 𝑁) = ∅)

Distinct variable groups:   𝑒,𝐺   𝑒,𝑁

Proof of Theorem uhgrnbgr0nb

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2763	. . . . . 6 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	nbuhgr 29981	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
4	3	adantlr 728	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
5		df-nel 3066	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∉ 𝑒 ↔ ¬ 𝑁 ∈ 𝑒)
6		prssg 4780	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
7		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → 𝑁 ∈ 𝑒)
8	6, 7	biimtrrdi 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑒))
9	8	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑒))
10	9	con3d 153	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (¬ 𝑁 ∈ 𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
11	5, 10	biimtrid 245	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑁 ∉ 𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
12	11	ralimdva 3178	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) → (∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒 → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒))
13	12	imp 412	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒)
14		ralnex 3092	. . . . . . . . . . 11 ⊢ (∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒 ↔ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
15	13, 14	sylib 221	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
16	15	expcom 419	. . . . . . . . 9 ⊢ (∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒 → ((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
17	16	expd 421	. . . . . . . 8 ⊢ (∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒 → (𝐺 ∈ UHGraph → ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)))
18	17	impcom 413	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
19	18	expdimp 458	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → (𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
20	19	ralrimiv 3157	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
21		rabeq0 4341	. . . . 5 ⊢ ({𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
22	20, 21	sylibr 237	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = ∅)
23	4, 22	eqtrd 2798	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
24	23	expcom 419	. 2 ⊢ (𝑁 ∈ V → ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → (𝐺 NeighbVtx 𝑁) = ∅))
25		id 23	. . . . 5 ⊢ (¬ 𝑁 ∈ V → ¬ 𝑁 ∈ V)
26	25	intnand 494	. . . 4 ⊢ (¬ 𝑁 ∈ V → ¬ (𝐺 ∈ V ∧ 𝑁 ∈ V))
27		nbgrprc0 29972	. . . 4 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
28	26, 27	syl 18	. . 3 ⊢ (¬ 𝑁 ∈ V → (𝐺 NeighbVtx 𝑁) = ∅)
29	28	a1d 26	. 2 ⊢ (¬ 𝑁 ∈ V → ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → (𝐺 NeighbVtx 𝑁) = ∅))
30	24, 29	pm2.61i 184	1 ⊢ ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → (𝐺 NeighbVtx 𝑁) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ∉ wnel 3065  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  {csn 4588  {cpr 4590  ‘cfv 6566  (class class class)co 7451  Vtxcvtx 29631  Edgcedg 29682  UHGraphcuhgr 29691   NeighbVtx cnbgr 29970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-edg 29683  df-uhgr 29693  df-nbgr 29971

This theorem is used by: (None)