Theorem cusgrusgr 29445

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

Step	Hyp	Ref	Expression
1		iscusgr 29444	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2103  USGraphcusgr 29175  ComplGraphccplgr 29435  ComplUSGraphccusgr 29436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3484  df-in 3977  df-cusgr 29438

This theorem is referenced by:  cusgrres  29475  cusgrsizeindslem  29478  cusgrsizeinds  29479  cusgrsize  29481  cusgrrusgr  29608  cusgredgex  35081  cusgr3cyclex  35096  cusgracyclt3v  35116