Theorem cusgrusgr 29462

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 29461	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2	1	simplbi 497	1 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2108  USGraphcusgr 29192  ComplGraphccplgr 29452  ComplUSGraphccusgr 29453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3483  df-in 3973  df-cusgr 29455

This theorem is referenced by:  cusgrres  29492  cusgrsizeindslem  29495  cusgrsizeinds  29496  cusgrsize  29498  cusgrrusgr  29625  cusgredgex  35120  cusgr3cyclex  35134  cusgracyclt3v  35154