Theorem uhgrnbgr0nb 29512

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 2761	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2761	. . . . . 6 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	nbuhgr 29501	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
4	3	adantlr 725	. . . 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 3061	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∉ 𝑒 ↔ ¬ 𝑁 ∈ 𝑒)
6		prssg 4774	. . . . . . . . . . . . . . . . 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	biimtrrdi 256	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑒))
9	8	ad2antlr 737	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑒))
10	9	con3d 152	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (¬ 𝑁 ∈ 𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
11	5, 10	biimtrid 244	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑁 ∉ 𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
12	11	ralimdva 3173	. . . . . . . . . . . 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 3087	. . . . . . . . . . 11 ⊢ (∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒 ↔ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
15	13, 14	sylib 220	. . . . . . . . . 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 3152	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
21		rabeq0 4339	. . . . 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 236	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) ∧ 𝑁 ∈ V) → {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = ∅)
23	4, 22	eqtrd 2796	. . 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 29492	. . . 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 183	1 ⊢ ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → (𝐺 NeighbVtx 𝑁) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∉ wnel 3060  ∀wral 3075  ∃wrex 3085  {crab 3413  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4283  {csn 4579  {cpr 4581  ‘cfv 6516  (class class class)co 7391  Vtxcvtx 29154  Edgcedg 29205  UHGraphcuhgr 29214   NeighbVtx cnbgr 29490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-edg 29206  df-uhgr 29216  df-nbgr 29491

This theorem is referenced by: (None)