Theorem iscusgr 28429

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 28423	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4162	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  USGraphcusgr 28163  ComplGraphccplgr 28420  ComplUSGraphccusgr 28421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-in 3920  df-cusgr 28423

This theorem is referenced by:  cusgrusgr  28430  cusgrcplgr  28431  iscusgrvtx  28432  cusgruvtxb  28433  iscusgredg  28434  cusgr0  28437  cusgr0v  28439  cusgr1v  28442  cusgrop  28449  cusgrexi  28454  structtocusgr  28457  cusgrres  28459