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

Theorem predgclnbgrel 47712
Description: If a (not necessarily proper) unordered pair containing a vertex is an edge, the other vertex is in the closed neighborhood of the first vertex. (Contributed by AV, 23-Aug-2025.)
Hypotheses
Ref Expression
predgclnbgrel.v 𝑉 = (Vtx‘𝐺)
predgclnbgrel.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
predgclnbgrel ((𝑁𝑉𝑋𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋))

Proof of Theorem predgclnbgrel
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 3simpa 1146 . 2 ((𝑁𝑉𝑋𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → (𝑁𝑉𝑋𝑉))
2 simp3 1136 . . . 4 ((𝑁𝑉𝑋𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ∈ 𝐸)
3 sseq2 4022 . . . . 5 (𝑒 = {𝑋, 𝑁} → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁}))
43adantl 481 . . . 4 (((𝑁𝑉𝑋𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) ∧ 𝑒 = {𝑋, 𝑁}) → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁}))
5 ssidd 4019 . . . 4 ((𝑁𝑉𝑋𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ⊆ {𝑋, 𝑁})
62, 4, 5rspcedvd 3624 . . 3 ((𝑁𝑉𝑋𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)
76olcd 873 . 2 ((𝑁𝑉𝑋𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → (𝑁 = 𝑋 ∨ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
8 predgclnbgrel.v . . 3 𝑉 = (Vtx‘𝐺)
9 predgclnbgrel.e . . 3 𝐸 = (Edg‘𝐺)
108, 9clnbgrel 47702 . 2 (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
111, 7, 10sylanbrc 582 1 ((𝑁𝑉𝑋𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 846  w3a 1085   = wceq 1535  wcel 2104  wrex 3066  wss 3963  {cpr 4632  cfv 6558  (class class class)co 7425  Vtxcvtx 29009  Edgcedg 29060   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-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-1st 8007  df-2nd 8008  df-clnbgr 47694
This theorem is referenced by:  grlimgrtri  47821
  Copyright terms: Public domain W3C validator