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Theorem uhgredgn0 26910
 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 26831 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 eqid 2824 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2824 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3uhgrf 26844 . . . 4 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
54frnd 6502 . . 3 (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
61, 5eqsstrid 3999 . 2 (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
76sselda 3951 1 ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2115   ∖ cdif 3915  ∅c0 4274  𝒫 cpw 4520  {csn 4548  dom cdm 5536  ran crn 5537  ‘cfv 6336  Vtxcvtx 26778  iEdgciedg 26779  Edgcedg 26829  UHGraphcuhgr 26838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-edg 26830  df-uhgr 26840 This theorem is referenced by:  edguhgr  26911  uhgredgss  26913  uhgrvd00  27313  lfuhgr2  32383  loop1cycl  32402
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