Theorem iscusgr 30004

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 29998	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4100	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  USGraphcusgr 29735  ComplGraphccplgr 29995  ComplUSGraphccusgr 29996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-cusgr 29998

This theorem is used by:  cusgrusgr  30005  cusgrcplgr  30006  iscusgrvtx  30007  cusgruvtxb  30008  iscusgredg  30009  cusgr0  30012  cusgr0v  30014  cusgr1v  30017  cusgrop  30024  cusgrexi  30029  structtocusgr  30032  cusgrres  30034