Theorem iscusgr 29398

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 29392	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4152	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  USGraphcusgr 29129  ComplGraphccplgr 29389  ComplUSGraphccusgr 29390

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 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-in 3905  df-cusgr 29392

This theorem is referenced by:  cusgrusgr  29399  cusgrcplgr  29400  iscusgrvtx  29401  cusgruvtxb  29402  iscusgredg  29403  cusgr0  29406  cusgr0v  29408  cusgr1v  29411  cusgrop  29418  cusgrexi  29423  structtocusgr  29426  cusgrres  29429