Theorem uvtxisvtx 26018

Description: A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)

Assertion

Ref	Expression
uvtxisvtx	⊢ (𝑁 ∈ (𝑉 UnivVertex 𝐸) → 𝑁 ∈ 𝑉)

Proof of Theorem uvtxisvtx

Dummy variables 𝑒 𝑘 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isuvtx 26016	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UnivVertex 𝐸) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})
2	1	eleq2d 2673	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸}))
3		elrabi 3328	. . 3 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸} → 𝑁 ∈ 𝑉)
4	2, 3	syl6bi 242	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) → 𝑁 ∈ 𝑉))
5		df-uvtx 25951	. . . 4 ⊢ UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛 ∈ 𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒})
6	5	mpt2ndm0 6773	. . 3 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UnivVertex 𝐸) = ∅)
7		n0i 3879	. . . 4 ⊢ (𝑁 ∈ (𝑉 UnivVertex 𝐸) → ¬ (𝑉 UnivVertex 𝐸) = ∅)
8	7	pm2.21d 117	. . 3 ⊢ (𝑁 ∈ (𝑉 UnivVertex 𝐸) → ((𝑉 UnivVertex 𝐸) = ∅ → 𝑁 ∈ 𝑉))
9	6, 8	syl5com 31	. 2 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) → 𝑁 ∈ 𝑉))
10	4, 9	pm2.61i 175	1 ⊢ (𝑁 ∈ (𝑉 UnivVertex 𝐸) → 𝑁 ∈ 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {csn 4125  {cpr 4127  ran crn 5039  (class class class)co 6549   UnivVertex cuvtx 25948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-uvtx 25951

This theorem is referenced by:  uvtxnm1nbgra  26022  vdiscusgra  26448