Theorem iscusgr 29491

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 29485	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4155	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  USGraphcusgr 29222  ComplGraphccplgr 29482  ComplUSGraphccusgr 29483

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-in 3908  df-cusgr 29485

This theorem is referenced by:  cusgrusgr  29492  cusgrcplgr  29493  iscusgrvtx  29494  cusgruvtxb  29495  iscusgredg  29496  cusgr0  29499  cusgr0v  29501  cusgr1v  29504  cusgrop  29511  cusgrexi  29516  structtocusgr  29519  cusgrres  29522