Theorem cusgrcplgr 27690

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 27688	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2	1	simprbi 496	1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)

Syntax hints:   → wi 4   ∈ wcel 2108  USGraphcusgr 27422  ComplGraphccplgr 27679  ComplUSGraphccusgr 27680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-cusgr 27682

This theorem is referenced by:  cusgrsizeindslem  27721  cusgrrusgr  27851  cusgredgex  32983