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Theorem uhgr0vusgr 26589
Description: The null graph, with no vertices, represented by a hypergraph, is a simple graph. (Contributed by AV, 5-Dec-2020.)
Assertion
Ref Expression
uhgr0vusgr ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph)

Proof of Theorem uhgr0vusgr
StepHypRef Expression
1 simpl 476 . 2 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ UHGraph)
2 eqid 2778 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2778 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
42, 3uhgr0v0e 26585 . . 3 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (Edg‘𝐺) = ∅)
5 uhgriedg0edg0 26475 . . . 4 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
65adantr 474 . . 3 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
74, 6mpbid 224 . 2 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
81, 7usgr0e 26583 1 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  c0 4141  cfv 6135  Vtxcvtx 26344  iEdgciedg 26345  Edgcedg 26395  UHGraphcuhgr 26404  USGraphcusgr 26498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fv 6143  df-edg 26396  df-uhgr 26406  df-usgr 26500
This theorem is referenced by: (None)
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