MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgr0vusgr Structured version   Visualization version   GIF version

Theorem uhgr0vusgr 29259
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 482 . 2 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ UHGraph)
2 eqid 2737 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2737 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
42, 3uhgr0v0e 29255 . . 3 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (Edg‘𝐺) = ∅)
5 uhgriedg0edg0 29144 . . . 4 (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
65adantr 480 . . 3 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))
74, 6mpbid 232 . 2 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
81, 7usgr0e 29253 1 ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  c0 4333  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073  USGraphcusgr 29166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-edg 29065  df-uhgr 29075  df-usgr 29168
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator