Theorem iscusgr 26314

Description: The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)

Assertion

Ref	Expression
iscusgr	|- ( G e. ComplUSGraph <-> ( G e. USGraph /\ G e. ComplGraph ) )

Proof of Theorem iscusgr

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. 2 |- ( g = G -> ( g e. ComplGraph <-> G e. ComplGraph ) )
2		df-cusgr 26232	. 2 |- ComplUSGraph = { g e. USGraph | g e. ComplGraph }
3	1, 2	elrab2 3366	1 |- ( G e. ComplUSGraph <-> ( G e. USGraph /\ G e. ComplGraph ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   USGraph cusgr 26044  ComplGraphccplgr 26226  ComplUSGraphccusgr 26227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-cusgr 26232

This theorem is referenced by:  cusgrusgr  26315  cusgrcplgr  26316  iscusgrvtx  26317  cusgruvtxb  26318  iscusgredg  26319  cusgr0  26322  cusgr0v  26324  cusgr1v  26327  cusgrop  26334  cusgrexi  26339  structtocusgr  26342  cusgrres  26344