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Theorem umgra0 25854
 Description: The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
umgra0 (𝑉𝑊𝑉 UMGrph ∅)

Proof of Theorem umgra0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 f0 5999 . . 3 ∅:∅⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
2 dm0 5260 . . . 4 dom ∅ = ∅
32feq2i 5950 . . 3 (∅:dom ∅⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ ∅:∅⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
41, 3mpbir 220 . 2 ∅:dom ∅⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
5 0ex 4718 . . 3 ∅ ∈ V
6 isumgra 25844 . . 3 ((𝑉𝑊 ∧ ∅ ∈ V) → (𝑉 UMGrph ∅ ↔ ∅:dom ∅⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
75, 6mpan2 703 . 2 (𝑉𝑊 → (𝑉 UMGrph ∅ ↔ ∅:dom ∅⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
84, 7mpbiri 247 1 (𝑉𝑊𝑉 UMGrph ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979   UMGrph cumg 25841 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-umgra 25842 This theorem is referenced by:  eupa0  26501
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