Theorem clnbgrvtxel 48305

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 2737	. . 3 ⊢ (𝐾 ∈ 𝑉 → 𝐾 = 𝐾)
3	2	orcd 874	. 2 ⊢ (𝐾 ∈ 𝑉 → (𝐾 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝐾} ⊆ 𝑒))
4		clnbgrvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
5		eqid 2736	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
6	4, 5	clnbgrel 48304	. 2 ⊢ (𝐾 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ ((𝐾 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ (𝐾 = 𝐾 ∨ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝐾} ⊆ 𝑒)))
7	1, 1, 3, 6	syl21anbrc 1346	1 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ (𝐺 ClNeighbVtx 𝐾))

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ⊆ wss 3889  {cpr 4569  ‘cfv 6498  (class class class)co 7367  Vtxcvtx 29065  Edgcedg 29116   ClNeighbVtx cclnbgr 48294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-clnbgr 48295

This theorem is referenced by:  clnbgrn0  48308  clnbgrgrim  48410  isubgr3stgrlem7  48448  grlimprclnbgr  48472  grlimprclnbgrvtx  48475  grlimgredgex  48476  grlimgrtri  48479