Theorem iscusgr 28706

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 28700	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4198	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  USGraphcusgr 28440  ComplGraphccplgr 28697  ComplUSGraphccusgr 28698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-cusgr 28700

This theorem is referenced by:  cusgrusgr  28707  cusgrcplgr  28708  iscusgrvtx  28709  cusgruvtxb  28710  iscusgredg  28711  cusgr0  28714  cusgr0v  28716  cusgr1v  28719  cusgrop  28726  cusgrexi  28731  structtocusgr  28734  cusgrres  28736