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Theorem uvtx0 26019
 Description: There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
uvtx0 (∅ UnivVertex 𝐸) = ∅

Proof of Theorem uvtx0
Dummy variables 𝑒 𝑘 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isuvtx 26016 . . 3 ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ UnivVertex 𝐸) = {𝑛 ∈ ∅ ∣ ∀𝑘 ∈ (∅ ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})
2 ral0 4028 . . . . 5 𝑛 ∈ ∅ ¬ ∀𝑘 ∈ (∅ ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸
32a1i 11 . . . 4 ((∅ ∈ V ∧ 𝐸 ∈ V) → ∀𝑛 ∈ ∅ ¬ ∀𝑘 ∈ (∅ ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸)
4 rabeq0 3911 . . . 4 ({𝑛 ∈ ∅ ∣ ∀𝑘 ∈ (∅ ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸} = ∅ ↔ ∀𝑛 ∈ ∅ ¬ ∀𝑘 ∈ (∅ ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸)
53, 4sylibr 223 . . 3 ((∅ ∈ V ∧ 𝐸 ∈ V) → {𝑛 ∈ ∅ ∣ ∀𝑘 ∈ (∅ ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸} = ∅)
61, 5eqtrd 2644 . 2 ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ UnivVertex 𝐸) = ∅)
7 df-uvtx 25951 . . 3 UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒})
87mpt2ndm0 6773 . 2 (¬ (∅ ∈ V ∧ 𝐸 ∈ V) → (∅ UnivVertex 𝐸) = ∅)
96, 8pm2.61i 175 1 (∅ UnivVertex 𝐸) = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ 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: (None)
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