Theorem iscusgr 29981

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 29975	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4149	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  USGraphcusgr 29712  ComplGraphccplgr 29972  ComplUSGraphccusgr 29973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-in 3906  df-cusgr 29975

This theorem is used by:  cusgrusgr  29982  cusgrcplgr  29983  iscusgrvtx  29984  cusgruvtxb  29985  iscusgredg  29986  cusgr0  29989  cusgr0v  29991  cusgr1v  29994  cusgrop  30001  cusgrexi  30006  structtocusgr  30009  cusgrres  30011