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Theorem uhgrnbgr0nb 29371
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)𝑁𝑒) → (𝐺 NeighbVtx 𝑁) = ∅)
Distinct variable groups:   𝑒,𝐺   𝑒,𝑁

Proof of Theorem uhgrnbgr0nb
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2737 . . . . . 6 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbuhgr 29360 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
43adantlr 715 . . . 4 (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
5 df-nel 3047 . . . . . . . . . . . . . 14 (𝑁𝑒 ↔ ¬ 𝑁𝑒)
6 prssg 4819 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
7 simpl 482 . . . . . . . . . . . . . . . . 17 ((𝑁𝑒𝑛𝑒) → 𝑁𝑒)
86, 7biimtrrdi 254 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑒))
98ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑒))
109con3d 152 . . . . . . . . . . . . . 14 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (¬ 𝑁𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
115, 10biimtrid 242 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑁𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
1211ralimdva 3167 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) → (∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒 → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒))
1312imp 406 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒)
14 ralnex 3072 . . . . . . . . . . 11 (∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒 ↔ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
1513, 14sylib 218 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
1615expcom 413 . . . . . . . . 9 (∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒 → ((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
1716expd 415 . . . . . . . 8 (∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒 → (𝐺 ∈ UHGraph → ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)))
1817impcom 407 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
1918expdimp 452 . . . . . 6 (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) ∧ 𝑁 ∈ V) → (𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
2019ralrimiv 3145 . . . . 5 (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) ∧ 𝑁 ∈ V) → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
21 rabeq0 4388 . . . . 5 ({𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
2220, 21sylibr 234 . . . 4 (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) ∧ 𝑁 ∈ V) → {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = ∅)
234, 22eqtrd 2777 . . 3 (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
2423expcom 413 . 2 (𝑁 ∈ V → ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → (𝐺 NeighbVtx 𝑁) = ∅))
25 id 22 . . . . 5 𝑁 ∈ V → ¬ 𝑁 ∈ V)
2625intnand 488 . . . 4 𝑁 ∈ V → ¬ (𝐺 ∈ V ∧ 𝑁 ∈ V))
27 nbgrprc0 29351 . . . 4 (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
2826, 27syl 17 . . 3 𝑁 ∈ V → (𝐺 NeighbVtx 𝑁) = ∅)
2928a1d 25 . 2 𝑁 ∈ V → ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → (𝐺 NeighbVtx 𝑁) = ∅))
3024, 29pm2.61i 182 1 ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → (𝐺 NeighbVtx 𝑁) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wnel 3046  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  wss 3951  c0 4333  {csn 4626  {cpr 4628  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  Edgcedg 29064  UHGraphcuhgr 29073   NeighbVtx cnbgr 29349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-edg 29065  df-uhgr 29075  df-nbgr 29350
This theorem is referenced by: (None)
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