Theorem cusgrusgr 30000

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 29999	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2	1	simplbi 502	1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  USGraphcusgr 29730  ComplGraphccplgr 29990  ComplUSGraphccusgr 29991

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-cusgr 29993

This theorem is used by:  cusgrres  30029  cusgrsizeindslem  30032  cusgrsizeinds  30033  cusgrsize  30035  cusgrrusgr  30162  cusgredgex  35948  cusgr3cyclex  35953  cusgracyclt3v  35963