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Theorem relusgra 25864
 Description: The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
relusgra Rel USGrph

Proof of Theorem relusgra
Dummy variables 𝑣 𝑒 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-usgra 25862 . 2 USGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}
21relopabi 5167 1 Rel USGrph
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  dom cdm 5038  Rel wrel 5043  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  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-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-usgra 25862 This theorem is referenced by:  usgrav  25867  elusuhgra  25897  usgrafis  25944
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