Theorem cusgrusgr 29780

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 29779	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2	1	simplbi 501	1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  USGraphcusgr 29510  ComplGraphccplgr 29770  ComplUSGraphccusgr 29771

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-cusgr 29773

This theorem is used by:  cusgrres  29809  cusgrsizeindslem  29812  cusgrsizeinds  29813  cusgrsize  29815  cusgrrusgr  29942  cusgredgex  35622  cusgr3cyclex  35636  cusgracyclt3v  35656