Theorem clnbgrvtxel 47803

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 2730	. . 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 2729	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
6	4, 5	clnbgrel 47802	. 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 3053   ⊆ wss 3911  {cpr 4587  ‘cfv 6499  (class class class)co 7369  Vtxcvtx 28899  Edgcedg 28950   ClNeighbVtx cclnbgr 47792

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 47793

This theorem is referenced by:  clnbgrn0  47806  clnbgrgrim  47907  isubgr3stgrlem7  47944  grlimgrtri  47968