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Theorem clnbgredg 47799
Description: A vertex connected by an edge with another vertex is a neighbor of that vertex. (Contributed by AV, 24-Aug-2025.)
Hypotheses
Ref Expression
clnbgredg.e 𝐸 = (Edg‘𝐺)
clnbgredg.n 𝑁 = (𝐺 ClNeighbVtx 𝑋)
Assertion
Ref Expression
clnbgredg ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → 𝑌𝑁)

Proof of Theorem clnbgredg
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 clnbgredg.e . . . . . . . . . 10 𝐸 = (Edg‘𝐺)
21eleq2i 2832 . . . . . . . . 9 (𝐾𝐸𝐾 ∈ (Edg‘𝐺))
32biimpi 216 . . . . . . . 8 (𝐾𝐸𝐾 ∈ (Edg‘𝐺))
433ad2ant1 1134 . . . . . . 7 ((𝐾𝐸𝑋𝐾𝑌𝐾) → 𝐾 ∈ (Edg‘𝐺))
5 simp3 1139 . . . . . . 7 ((𝐾𝐸𝑋𝐾𝑌𝐾) → 𝑌𝐾)
64, 5jca 511 . . . . . 6 ((𝐾𝐸𝑋𝐾𝑌𝐾) → (𝐾 ∈ (Edg‘𝐺) ∧ 𝑌𝐾))
76anim2i 617 . . . . 5 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → (𝐺 ∈ UHGraph ∧ (𝐾 ∈ (Edg‘𝐺) ∧ 𝑌𝐾)))
8 3anass 1095 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ (Edg‘𝐺) ∧ 𝑌𝐾) ↔ (𝐺 ∈ UHGraph ∧ (𝐾 ∈ (Edg‘𝐺) ∧ 𝑌𝐾)))
97, 8sylibr 234 . . . 4 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → (𝐺 ∈ UHGraph ∧ 𝐾 ∈ (Edg‘𝐺) ∧ 𝑌𝐾))
10 uhgredgrnv 29137 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ (Edg‘𝐺) ∧ 𝑌𝐾) → 𝑌 ∈ (Vtx‘𝐺))
119, 10syl 17 . . 3 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → 𝑌 ∈ (Vtx‘𝐺))
12 simp2 1138 . . . . . . 7 ((𝐾𝐸𝑋𝐾𝑌𝐾) → 𝑋𝐾)
134, 12jca 511 . . . . . 6 ((𝐾𝐸𝑋𝐾𝑌𝐾) → (𝐾 ∈ (Edg‘𝐺) ∧ 𝑋𝐾))
1413anim2i 617 . . . . 5 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → (𝐺 ∈ UHGraph ∧ (𝐾 ∈ (Edg‘𝐺) ∧ 𝑋𝐾)))
15 3anass 1095 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ (Edg‘𝐺) ∧ 𝑋𝐾) ↔ (𝐺 ∈ UHGraph ∧ (𝐾 ∈ (Edg‘𝐺) ∧ 𝑋𝐾)))
1614, 15sylibr 234 . . . 4 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → (𝐺 ∈ UHGraph ∧ 𝐾 ∈ (Edg‘𝐺) ∧ 𝑋𝐾))
17 uhgredgrnv 29137 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ (Edg‘𝐺) ∧ 𝑋𝐾) → 𝑋 ∈ (Vtx‘𝐺))
1816, 17syl 17 . . 3 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → 𝑋 ∈ (Vtx‘𝐺))
19 simpr1 1195 . . . . 5 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → 𝐾𝐸)
20 sseq2 4009 . . . . . 6 (𝑒 = 𝐾 → ({𝑋, 𝑌} ⊆ 𝑒 ↔ {𝑋, 𝑌} ⊆ 𝐾))
2120adantl 481 . . . . 5 (((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) ∧ 𝑒 = 𝐾) → ({𝑋, 𝑌} ⊆ 𝑒 ↔ {𝑋, 𝑌} ⊆ 𝐾))
22 prssi 4819 . . . . . . 7 ((𝑋𝐾𝑌𝐾) → {𝑋, 𝑌} ⊆ 𝐾)
23223adant1 1131 . . . . . 6 ((𝐾𝐸𝑋𝐾𝑌𝐾) → {𝑋, 𝑌} ⊆ 𝐾)
2423adantl 481 . . . . 5 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → {𝑋, 𝑌} ⊆ 𝐾)
2519, 21, 24rspcedvd 3623 . . . 4 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → ∃𝑒𝐸 {𝑋, 𝑌} ⊆ 𝑒)
2625olcd 875 . . 3 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → (𝑌 = 𝑋 ∨ ∃𝑒𝐸 {𝑋, 𝑌} ⊆ 𝑒))
27 eqid 2736 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2827, 1clnbgrel 47788 . . 3 (𝑌 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑌 ∈ (Vtx‘𝐺) ∧ 𝑋 ∈ (Vtx‘𝐺)) ∧ (𝑌 = 𝑋 ∨ ∃𝑒𝐸 {𝑋, 𝑌} ⊆ 𝑒)))
2911, 18, 26, 28syl21anbrc 1345 . 2 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → 𝑌 ∈ (𝐺 ClNeighbVtx 𝑋))
30 clnbgredg.n . . 3 𝑁 = (𝐺 ClNeighbVtx 𝑋)
3130eleq2i 2832 . 2 (𝑌𝑁𝑌 ∈ (𝐺 ClNeighbVtx 𝑋))
3229, 31sylibr 234 1 ((𝐺 ∈ UHGraph ∧ (𝐾𝐸𝑋𝐾𝑌𝐾)) → 𝑌𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wrex 3069  wss 3950  {cpr 4626  cfv 6559  (class class class)co 7429  Vtxcvtx 29003  Edgcedg 29054  UHGraphcuhgr 29063   ClNeighbVtx cclnbgr 47778
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 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751
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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  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 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-edg 29055  df-uhgr 29065  df-clnbgr 47779
This theorem is referenced by:  clnbgrssedg  47800  grlimgrtrilem1  47934
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