Theorem iscusgr 29397

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 29391	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4178	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  USGraphcusgr 29128  ComplGraphccplgr 29388  ComplUSGraphccusgr 29389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-in 3933  df-cusgr 29391

This theorem is referenced by:  cusgrusgr  29398  cusgrcplgr  29399  iscusgrvtx  29400  cusgruvtxb  29401  iscusgredg  29402  cusgr0  29405  cusgr0v  29407  cusgr1v  29410  cusgrop  29417  cusgrexi  29422  structtocusgr  29425  cusgrres  29428