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Theorem uhgredgn0 29146
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 29067 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 eqid 2736 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2736 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3uhgrf 29080 . . . 4 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
54frnd 6743 . . 3 (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
61, 5eqsstrid 4021 . 2 (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
76sselda 3982 1 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  cdif 3947  c0 4332  𝒫 cpw 4599  {csn 4625  dom cdm 5684  ran crn 5685  cfv 6560  Vtxcvtx 29014  iEdgciedg 29015  Edgcedg 29065  UHGraphcuhgr 29074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-edg 29066  df-uhgr 29076
This theorem is referenced by:  edguhgr  29147  uhgredgss  29149  uhgrvd00  29553  lfuhgr2  35125  loop1cycl  35143  uspgrimprop  47878
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