Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cusgrusgr Structured version   Visualization version   GIF version

Theorem cusgrusgr 40641
 Description: A complete simple graph is a simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
Assertion
Ref Expression
cusgrusgr (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph )

Proof of Theorem cusgrusgr
StepHypRef Expression
1 iscusgr 40640 . 2 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
21simplbi 475 1 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   USGraph cusgr 40379  ComplGraphccplgr 40552  ComplUSGraphccusgr 40553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-cusgr 40558 This theorem is referenced by:  cusgrres  40664  cusgrsizeindslem  40667  cusgrsizeinds  40668  cusgrsize  40670  cusgrrusgr  40781
 Copyright terms: Public domain W3C validator