Theorem iscusgr 29435

Description: The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)

Assertion

Ref	Expression
iscusgr	⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Proof of Theorem iscusgr

Step	Hyp	Ref	Expression
1		df-cusgr 29429	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4203	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  USGraphcusgr 29166  ComplGraphccplgr 29426  ComplUSGraphccusgr 29427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  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 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-cusgr 29429

This theorem is referenced by:  cusgrusgr  29436  cusgrcplgr  29437  iscusgrvtx  29438  cusgruvtxb  29439  iscusgredg  29440  cusgr0  29443  cusgr0v  29445  cusgr1v  29448  cusgrop  29455  cusgrexi  29460  structtocusgr  29463  cusgrres  29466