Theorem iscusgr 29453

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 29447	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4226	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  USGraphcusgr 29184  ComplGraphccplgr 29444  ComplUSGraphccusgr 29445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-in 3983  df-cusgr 29447

This theorem is referenced by:  cusgrusgr  29454  cusgrcplgr  29455  iscusgrvtx  29456  cusgruvtxb  29457  iscusgredg  29458  cusgr0  29461  cusgr0v  29463  cusgr1v  29466  cusgrop  29473  cusgrexi  29478  structtocusgr  29481  cusgrres  29484