Theorem iscusgr 29677

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 29671	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4158	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2145  USGraphcusgr 29408  ComplGraphccplgr 29668  ComplUSGraphccusgr 29669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3914  df-cusgr 29671

This theorem is referenced by:  cusgrusgr  29678  cusgrcplgr  29679  iscusgrvtx  29680  cusgruvtxb  29681  iscusgredg  29682  cusgr0  29685  cusgr0v  29687  cusgr1v  29690  cusgrop  29697  cusgrexi  29702  structtocusgr  29705  cusgrres  29707