Theorem iscusgr 29512

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 29506	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4139	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  USGraphcusgr 29243  ComplGraphccplgr 29503  ComplUSGraphccusgr 29504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-in 3897  df-cusgr 29506

This theorem is referenced by:  cusgrusgr  29513  cusgrcplgr  29514  iscusgrvtx  29515  cusgruvtxb  29516  iscusgredg  29517  cusgr0  29520  cusgr0v  29522  cusgr1v  29525  cusgrop  29532  cusgrexi  29537  structtocusgr  29540  cusgrres  29542