Theorem clnbgrvtxel 47834

Description: Every vertex 𝐾 is a member of its closed neighborhood. (Contributed by AV, 10-May-2025.)

Hypothesis

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

Assertion

Ref	Expression
clnbgrvtxel	⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ (𝐺 ClNeighbVtx 𝐾))

Proof of Theorem clnbgrvtxel

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ 𝑉)
2		eqidd 2731	. . 3 ⊢ (𝐾 ∈ 𝑉 → 𝐾 = 𝐾)
3	2	orcd 873	. 2 ⊢ (𝐾 ∈ 𝑉 → (𝐾 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝐾} ⊆ 𝑒))
4		clnbgrvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
5		eqid 2730	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
6	4, 5	clnbgrel 47833	. 2 ⊢ (𝐾 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ ((𝐾 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ (𝐾 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝐾} ⊆ 𝑒)))
7	1, 1, 3, 6	syl21anbrc 1345	1 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ (𝐺 ClNeighbVtx 𝐾))

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3054   ⊆ wss 3917  {cpr 4594  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  Edgcedg 28981   ClNeighbVtx cclnbgr 47823

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-clnbgr 47824

This theorem is referenced by:  clnbgrn0  47837  clnbgrgrim  47938  isubgr3stgrlem7  47975  grlimgrtri  47999