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Theorem uhgredgn0 28985
Description: An edge of a hypergraph is a nonempty subset of vertices. (Contributed by AV, 28-Nov-2020.)
Assertion
Ref Expression
uhgredgn0 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Proof of Theorem uhgredgn0
StepHypRef Expression
1 edgval 28906 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 eqid 2725 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2725 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3uhgrf 28919 . . . 4 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
54frnd 6725 . . 3 (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
61, 5eqsstrid 4021 . 2 (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
76sselda 3972 1 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  cdif 3936  c0 4318  𝒫 cpw 4598  {csn 4624  dom cdm 5672  ran crn 5673  cfv 6543  Vtxcvtx 28853  iEdgciedg 28854  Edgcedg 28904  UHGraphcuhgr 28913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-edg 28905  df-uhgr 28915
This theorem is referenced by:  edguhgr  28986  uhgredgss  28988  uhgrvd00  29392  lfuhgr2  34785  loop1cycl  34804  uspgrimprop  47283
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