Theorem iscusgr 27783

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 27777	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4136	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2110  USGraphcusgr 27517  ComplGraphccplgr 27774  ComplUSGraphccusgr 27775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-in 3899  df-cusgr 27777

This theorem is referenced by:  cusgrusgr  27784  cusgrcplgr  27785  iscusgrvtx  27786  cusgruvtxb  27787  iscusgredg  27788  cusgr0  27791  cusgr0v  27793  cusgr1v  27796  cusgrop  27803  cusgrexi  27808  structtocusgr  27811  cusgrres  27813