Theorem uhgrnbgr0nb 27144

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 2798	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2798	. . . . . 6 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	nbuhgr 27133	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
4	3	adantlr 714	. . . 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 3092	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∉ 𝑒 ↔ ¬ 𝑁 ∈ 𝑒)
6		prssg 4712	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
7		simpl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → 𝑁 ∈ 𝑒)
8	6, 7	syl6bir 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑒))
9	8	ad2antlr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑒))
10	9	con3d 155	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (¬ 𝑁 ∈ 𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
11	5, 10	syl5bi 245	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑁 ∉ 𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
12	11	ralimdva 3144	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) → (∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒 → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒))
13	12	imp 410	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒)
14		ralnex 3199	. . . . . . . . . . 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 417	. . . . . . . . 9 ⊢ (∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒 → ((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
17	16	expd 419	. . . . . . . 8 ⊢ (∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒 → (𝐺 ∈ UHGraph → ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)))
18	17	impcom 411	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
19	18	expdimp 456	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → (𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
20	19	ralrimiv 3148	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
21		rabeq0 4292	. . . . 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 2833	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
24	23	expcom 417	. 2 ⊢ (𝑁 ∈ V → ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → (𝐺 NeighbVtx 𝑁) = ∅))
25		id 22	. . . . 5 ⊢ (¬ 𝑁 ∈ V → ¬ 𝑁 ∈ V)
26	25	intnand 492	. . . 4 ⊢ (¬ 𝑁 ∈ V → ¬ (𝐺 ∈ V ∧ 𝑁 ∈ V))
27		nbgrprc0 27124	. . . 4 ⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
28	26, 27	syl 17	. . 3 ⊢ (¬ 𝑁 ∈ V → (𝐺 NeighbVtx 𝑁) = ∅)
29	28	a1d 25	. 2 ⊢ (¬ 𝑁 ∈ V → ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → (𝐺 NeighbVtx 𝑁) = ∅))
30	24, 29	pm2.61i 185	1 ⊢ ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → (𝐺 NeighbVtx 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∉ wnel 3091  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  {csn 4525  {cpr 4527  ‘cfv 6324  (class class class)co 7135  Vtxcvtx 26789  Edgcedg 26840  UHGraphcuhgr 26849   NeighbVtx cnbgr 27122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-edg 26841  df-uhgr 26851  df-nbgr 27123

This theorem is referenced by: (None)