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

Theorem iscusgr 27197
 Description: The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)
Assertion
Ref Expression
iscusgr (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Proof of Theorem iscusgr
StepHypRef Expression
1 df-cusgr 27191 . 2 ComplUSGraph = (USGraph ∩ ComplGraph)
21elin2 4157 1 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2115  USGraphcusgr 26931  ComplGraphccplgr 27188  ComplUSGraphccusgr 27189 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-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-in 3925  df-cusgr 27191 This theorem is referenced by:  cusgrusgr  27198  cusgrcplgr  27199  iscusgrvtx  27200  cusgruvtxb  27201  iscusgredg  27202  cusgr0  27205  cusgr0v  27207  cusgr1v  27210  cusgrop  27217  cusgrexi  27222  structtocusgr  27225  cusgrres  27227
 Copyright terms: Public domain W3C validator