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Theorem uhgriedg0edg0 26906
 Description: A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 8-Dec-2021.)
Assertion
Ref Expression
uhgriedg0edg0 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))

Proof of Theorem uhgriedg0edg0
StepHypRef Expression
1 eqid 2821 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21uhgrfun 26845 . 2 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3 eqid 2821 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
41, 3edg0iedg0 26834 . 2 (Fun (iEdg‘𝐺) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
52, 4syl 17 1 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∅c0 4291  Fun wfun 6344  ‘cfv 6350  iEdgciedg 26776  Edgcedg 26826  UHGraphcuhgr 26835 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-edg 26827  df-uhgr 26837 This theorem is referenced by:  uhgr0v0e  27014  uhgr0vusgr  27018  lfuhgr1v0e  27030  usgr1vr  27031  usgr1v0e  27102  uhgr0edg0rgr  27349  rgrusgrprc  27365
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