Theorem iscusgr 29705

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 29699	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4164	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  USGraphcusgr 29436  ComplGraphccplgr 29696  ComplUSGraphccusgr 29697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-in 3920  df-cusgr 29699

This theorem is referenced by:  cusgrusgr  29706  cusgrcplgr  29707  iscusgrvtx  29708  cusgruvtxb  29709  iscusgredg  29710  cusgr0  29713  cusgr0v  29715  cusgr1v  29718  cusgrop  29725  cusgrexi  29730  structtocusgr  29733  cusgrres  29735