Theorem clnbgr0vtx 47840

Description: In a null graph (with no vertices), all closed neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)

Assertion

Ref	Expression
clnbgr0vtx	⊢ ((Vtx‘𝐺) = ∅ → (𝐺 ClNeighbVtx 𝐾) = ∅)

Proof of Theorem clnbgr0vtx

Step	Hyp	Ref	Expression
1		nel02 4292	. . 3 ⊢ ((Vtx‘𝐺) = ∅ → ¬ 𝐾 ∈ (Vtx‘𝐺))
2		df-nel 3030	. . 3 ⊢ (𝐾 ∉ (Vtx‘𝐺) ↔ ¬ 𝐾 ∈ (Vtx‘𝐺))
3	1, 2	sylibr 234	. 2 ⊢ ((Vtx‘𝐺) = ∅ → 𝐾 ∉ (Vtx‘𝐺))
4		eqid 2729	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	clnbgrnvtx0 47831	. 2 ⊢ (𝐾 ∉ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝐾) = ∅)
6	3, 5	syl 17	1 ⊢ ((Vtx‘𝐺) = ∅ → (𝐺 ClNeighbVtx 𝐾) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∉ wnel 3029  ∅c0 4286  ‘cfv 6486  (class class class)co 7353  Vtxcvtx 28960   ClNeighbVtx cclnbgr 47822

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 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

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-nel 3030  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-clnbgr 47823

This theorem is referenced by: (None)