Theorem iscusgr 29565

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 29559	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4155	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  USGraphcusgr 29296  ComplGraphccplgr 29556  ComplUSGraphccusgr 29557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3911  df-cusgr 29559

This theorem is referenced by:  cusgrusgr  29566  cusgrcplgr  29567  iscusgrvtx  29568  cusgruvtxb  29569  iscusgredg  29570  cusgr0  29573  cusgr0v  29575  cusgr1v  29578  cusgrop  29585  cusgrexi  29590  structtocusgr  29593  cusgrres  29595