Theorem iscusgr 29308

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 29302	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4195	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∈ wcel 2098  USGraphcusgr 29039  ComplGraphccplgr 29299  ComplUSGraphccusgr 29300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-in 3951  df-cusgr 29302

This theorem is referenced by:  cusgrusgr  29309  cusgrcplgr  29310  iscusgrvtx  29311  cusgruvtxb  29312  iscusgredg  29313  cusgr0  29316  cusgr0v  29318  cusgr1v  29321  cusgrop  29328  cusgrexi  29333  structtocusgr  29336  cusgrres  29339