Theorem cusgrcplgr 29736

Description: A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.)

Assertion

Ref	Expression
cusgrcplgr	⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)

Proof of Theorem cusgrcplgr

Step	Hyp	Ref	Expression
1		iscusgr 29734	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2	1	simprbi 502	1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)

Syntax hints:   → wi 4   ∈ wcel 2141  USGraphcusgr 29465  ComplGraphccplgr 29725  ComplUSGraphccusgr 29726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3911  df-cusgr 29728

This theorem is referenced by:  cusgrsizeindslem  29767  cusgrrusgr  29897  cusgredgex  35568