Theorem iscusgr 29746

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 29740	. 2 ⊢ ComplUSGraph = (USGraph ∩ ComplGraph)
2	1	elin2 4157	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  USGraphcusgr 29477  ComplGraphccplgr 29737  ComplUSGraphccusgr 29738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3913  df-cusgr 29740

This theorem is referenced by:  cusgrusgr  29747  cusgrcplgr  29748  iscusgrvtx  29749  cusgruvtxb  29750  iscusgredg  29751  cusgr0  29754  cusgr0v  29756  cusgr1v  29759  cusgrop  29766  cusgrexi  29771  structtocusgr  29774  cusgrres  29776