Theorem clnbgrisvtx 47824

Description: Every member 𝑁 of the closed neighborhood of a vertex 𝐾 is a vertex. (Contributed by AV, 9-May-2025.)

Hypothesis

Ref	Expression
clnbgrvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
clnbgrisvtx	⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) → 𝑁 ∈ 𝑉)

Proof of Theorem clnbgrisvtx

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2729	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	clnbgrel 47822	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ ((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒)))
4		simpll 766	. 2 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒)) → 𝑁 ∈ 𝑉)
5	3, 4	sylbi 217	1 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) → 𝑁 ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3911  {cpr 4587  ‘cfv 6499  (class class class)co 7369  Vtxcvtx 28976  Edgcedg 29027   ClNeighbVtx cclnbgr 47812

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-clnbgr 47813

This theorem is referenced by:  clnbgrssvtx  47825  clnbgrgrim  47927  grlimgrtri  47988