Theorem iscusgr 27688

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 27682	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4127	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  USGraphcusgr 27422  ComplGraphccplgr 27679  ComplUSGraphccusgr 27680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-cusgr 27682

This theorem is referenced by:  cusgrusgr  27689  cusgrcplgr  27690  iscusgrvtx  27691  cusgruvtxb  27692  iscusgredg  27693  cusgr0  27696  cusgr0v  27698  cusgr1v  27701  cusgrop  27708  cusgrexi  27713  structtocusgr  27716  cusgrres  27718