MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbgrisvtxOLD Structured version   Visualization version   GIF version

Theorem nbgrisvtxOLD 26459
Description: Obsolete version of nbgrisvtx 26457 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
nbgrisvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrisvtxOLD ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁𝑉)

Proof of Theorem nbgrisvtxOLD
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 nbgrisvtx.v . . . 4 𝑉 = (Vtx‘𝐺)
2 eqid 2770 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrelOLD 26456 . . 3 (𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒)))
4 simp1l 1238 . . 3 (((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) → 𝑁𝑉)
53, 4syl6bi 243 . 2 (𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁𝑉))
65imp 393 1 ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wrex 3061  wss 3721  {cpr 4316  cfv 6031  (class class class)co 6792  Vtxcvtx 26094  Edgcedg 26159   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-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-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-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-nbgr 26447
This theorem is referenced by:  nbgrssvtxOLD  26460  nbgrnself2OLD  26481  nbgrssovtxOLD  26482
  Copyright terms: Public domain W3C validator