Theorem iscusgr 27785

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 27779	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4131	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  USGraphcusgr 27519  ComplGraphccplgr 27776  ComplUSGraphccusgr 27777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-cusgr 27779

This theorem is referenced by:  cusgrusgr  27786  cusgrcplgr  27787  iscusgrvtx  27788  cusgruvtxb  27789  iscusgredg  27790  cusgr0  27793  cusgr0v  27795  cusgr1v  27798  cusgrop  27805  cusgrexi  27810  structtocusgr  27813  cusgrres  27815