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Theorem uhgrnbgr0nb 26472
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 2770 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2770 . . . . . 6 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbuhgr 26461 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
43adantlr 686 . . . 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 3046 . . . . . . . . . . . . . 14 (𝑁𝑒 ↔ ¬ 𝑁𝑒)
6 prssg 4483 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
7 simpl 468 . . . . . . . . . . . . . . . . 17 ((𝑁𝑒𝑛𝑒) → 𝑁𝑒)
86, 7syl6bir 244 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑒))
98ad2antlr 698 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑒))
109con3d 149 . . . . . . . . . . . . . 14 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (¬ 𝑁𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
115, 10syl5bi 232 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑁𝑒 → ¬ {𝑁, 𝑛} ⊆ 𝑒))
1211ralimdva 3110 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) → (∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒 → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒))
1312imp 393 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → ∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒)
14 ralnex 3140 . . . . . . . . . . 11 (∀𝑒 ∈ (Edg‘𝐺) ¬ {𝑁, 𝑛} ⊆ 𝑒 ↔ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
1513, 14sylib 208 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
1615expcom 398 . . . . . . . . 9 (∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒 → ((𝐺 ∈ UHGraph ∧ (𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}))) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
1716expd 400 . . . . . . . 8 (∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒 → (𝐺 ∈ UHGraph → ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)))
1817impcom 394 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → ((𝑁 ∈ V ∧ 𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁})) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
1918expdimp 440 . . . . . 6 (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) ∧ 𝑁 ∈ V) → (𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) → ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒))
2019ralrimiv 3113 . . . . 5 (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) ∧ 𝑁 ∈ V) → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
21 rabeq0 4101 . . . . 5 ({𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒)
2220, 21sylibr 224 . . . 4 (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) ∧ 𝑁 ∈ V) → {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = ∅)
234, 22eqtrd 2804 . . 3 (((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅)
2423expcom 398 . 2 (𝑁 ∈ V → ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → (𝐺 NeighbVtx 𝑁) = ∅))
25 id 22 . . . . 5 𝑁 ∈ V → ¬ 𝑁 ∈ V)
2625intnand 998 . . . 4 𝑁 ∈ V → ¬ (𝐺 ∈ V ∧ 𝑁 ∈ V))
27 nbgrprc0 26448 . . . 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 176 1 ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁𝑒) → (𝐺 NeighbVtx 𝑁) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1630  wcel 2144  wnel 3045  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  cdif 3718  wss 3721  c0 4061  {csn 4314  {cpr 4316  cfv 6031  (class class class)co 6792  Vtxcvtx 26094  Edgcedg 26159  UHGraphcuhgr 26171   NeighbVtx cnbgr 26446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-edg 26160  df-uhgr 26173  df-nbgr 26447
This theorem is referenced by: (None)
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