Theorem uvtxnbgrss 29906

Description: A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)

Hypothesis

Ref	Expression
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxnbgrss	⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑉 ∖ {𝑁}) ⊆ (𝐺 NeighbVtx 𝑁))

Proof of Theorem uvtxnbgrss

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	vtxnbuvtx 29905	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
3		dfss3 3919	. 2 ⊢ ((𝑉 ∖ {𝑁}) ⊆ (𝐺 NeighbVtx 𝑁) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
4	2, 3	sylibr 237	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑉 ∖ {𝑁}) ⊆ (𝐺 NeighbVtx 𝑁))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∖ cdif 3895   ⊆ wss 3898  {csn 4583  ‘cfv 6527  (class class class)co 7408  Vtxcvtx 29507   NeighbVtx cnbgr 29846  UnivVtxcuvtx 29899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6483  df-fun 6529  df-fv 6535  df-ov 7411  df-uvtx 29900

This theorem is used by:  uvtxnbgr  29914