Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clnbgrssedg Structured version   Visualization version   GIF version

Theorem clnbgrssedg 47714
Description: The vertices connected by an edge are a subset of the neighborhood of each of these vertices. (Contributed by AV, 26-May-2025.) (Proof shortened by AV, 24-Aug-2025.)
Hypotheses
Ref Expression
clnbgrssedg.e 𝐸 = (Edg‘𝐺)
clnbgrssedg.n 𝑁 = (𝐺 ClNeighbVtx 𝑋)
Assertion
Ref Expression
clnbgrssedg ((𝐺 ∈ UHGraph ∧ 𝐾𝐸𝑋𝐾) → 𝐾𝑁)

Proof of Theorem clnbgrssedg
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 clnbgrssedg.e . . . . 5 𝐸 = (Edg‘𝐺)
2 clnbgrssedg.n . . . . 5 𝑁 = (𝐺 ClNeighbVtx 𝑋)
31, 2clnbgredg 47713 . . . 4 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑣𝐾)) → 𝑣𝑁)
433exp2 1352 . . 3 (𝐺 ∈ UHGraph → (𝐾𝐸 → (𝑋𝐾 → (𝑣𝐾𝑣𝑁))))
543imp 1109 . 2 ((𝐺 ∈ UHGraph ∧ 𝐾𝐸𝑋𝐾) → (𝑣𝐾𝑣𝑁))
65ssrdv 4001 1 ((𝐺 ∈ UHGraph ∧ 𝐾𝐸𝑋𝐾) → 𝐾𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1535  wcel 2104  wss 3963  cfv 6558  (class class class)co 7425  Edgcedg 29060  UHGraphcuhgr 29069   ClNeighbVtx cclnbgr 47693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430  ax-un 7747
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-1st 8007  df-2nd 8008  df-edg 29061  df-uhgr 29071  df-clnbgr 47694
This theorem is referenced by:  grlimgrtrilem1  47819
  Copyright terms: Public domain W3C validator