Theorem iscusgr 26766

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 26760	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4024	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 198   ∧ wa 386   ∈ wcel 2107  USGraphcusgr 26498  ComplGraphccplgr 26757  ComplUSGraphccusgr 26758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-in 3799  df-cusgr 26760

This theorem is referenced by:  cusgrusgr  26767  cusgrcplgr  26768  iscusgrvtx  26769  cusgruvtxb  26770  iscusgredg  26771  cusgr0  26774  cusgr0v  26776  cusgr1v  26779  cusgrop  26786  cusgrexi  26791  structtocusgr  26794  cusgrres  26796