Theorem iscusgr 30032

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 30026	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4103	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  USGraphcusgr 29763  ComplGraphccplgr 30023  ComplUSGraphccusgr 30024

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-cusgr 30026

This theorem is used by:  cusgrusgr  30033  cusgrcplgr  30034  iscusgrvtx  30035  cusgruvtxb  30036  iscusgredg  30037  cusgr0  30040  cusgr0v  30042  cusgr1v  30045  cusgrop  30052  cusgrexi  30057  structtocusgr  30060  cusgrres  30062