Theorem cusgrusgr 27835

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 27834	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2	1	simplbi 499	1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2104  USGraphcusgr 27568  ComplGraphccplgr 27825  ComplUSGraphccusgr 27826

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-in 3899  df-cusgr 27828

This theorem is referenced by:  cusgrres  27864  cusgrsizeindslem  27867  cusgrsizeinds  27868  cusgrsize  27870  cusgrrusgr  27997  cusgredgex  33132  cusgr3cyclex  33147  cusgracyclt3v  33167