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Theorem uhgra0v 25839
 Description: The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
Assertion
Ref Expression
uhgra0v (∅ UHGrph 𝐸𝐸 = ∅)

Proof of Theorem uhgra0v
StepHypRef Expression
1 uhgrav 25825 . . 3 (∅ UHGrph 𝐸 → (∅ ∈ V ∧ 𝐸 ∈ V))
2 isuhgra 25827 . . . 4 ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 ∅ ∖ {∅})))
3 eqid 2610 . . . . . 6 dom 𝐸 = dom 𝐸
4 pw0 4283 . . . . . . . 8 𝒫 ∅ = {∅}
54difeq1i 3686 . . . . . . 7 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
6 difid 3902 . . . . . . 7 ({∅} ∖ {∅}) = ∅
75, 6eqtri 2632 . . . . . 6 (𝒫 ∅ ∖ {∅}) = ∅
83, 7feq23i 5952 . . . . 5 (𝐸:dom 𝐸⟶(𝒫 ∅ ∖ {∅}) ↔ 𝐸:dom 𝐸⟶∅)
9 f00 6000 . . . . . 6 (𝐸:dom 𝐸⟶∅ ↔ (𝐸 = ∅ ∧ dom 𝐸 = ∅))
109simplbi 475 . . . . 5 (𝐸:dom 𝐸⟶∅ → 𝐸 = ∅)
118, 10sylbi 206 . . . 4 (𝐸:dom 𝐸⟶(𝒫 ∅ ∖ {∅}) → 𝐸 = ∅)
122, 11syl6bi 242 . . 3 ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ UHGrph 𝐸𝐸 = ∅))
131, 12mpcom 37 . 2 (∅ UHGrph 𝐸𝐸 = ∅)
14 0ex 4718 . . . 4 ∅ ∈ V
15 uhgra0 25838 . . . 4 (∅ ∈ V → ∅ UHGrph ∅)
1614, 15ax-mp 5 . . 3 ∅ UHGrph ∅
17 breq2 4587 . . 3 (𝐸 = ∅ → (∅ UHGrph 𝐸 ↔ ∅ UHGrph ∅))
1816, 17mpbiri 247 . 2 (𝐸 = ∅ → ∅ UHGrph 𝐸)
1913, 18impbii 198 1 (∅ UHGrph 𝐸𝐸 = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶wf 5800   UHGrph cuhg 25819 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  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-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  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-fun 5806  df-fn 5807  df-f 5808  df-uhgra 25821 This theorem is referenced by: (None)
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