Theorem iscusgr 28943

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 28937	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4197	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2105  USGraphcusgr 28677  ComplGraphccplgr 28934  ComplUSGraphccusgr 28935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-cusgr 28937

This theorem is referenced by:  cusgrusgr  28944  cusgrcplgr  28945  iscusgrvtx  28946  cusgruvtxb  28947  iscusgredg  28948  cusgr0  28951  cusgr0v  28953  cusgr1v  28956  cusgrop  28963  cusgrexi  28968  structtocusgr  28971  cusgrres  28973