Theorem iscusgrvtx 26317

Description: A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.)

Hypothesis

Ref	Expression
iscusgrvtx.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
iscusgrvtx	|- ( G e. ComplUSGraph <-> ( G e. USGraph /\ A. v e. V v e. (UnivVtx ` G ) ) )

Distinct variable groups:   v, G   v, V

Proof of Theorem iscusgrvtx

Step	Hyp	Ref	Expression
1		iscusgr 26314	. 2 |- ( G e. ComplUSGraph <-> ( G e. USGraph /\ G e. ComplGraph ) )
2		iscusgrvtx.v	. . . 4 |- V = (Vtx ` G )
3	2	iscplgr 26310	. . 3 |- ( G e. USGraph -> ( G e. ComplGraph <-> A. v e. V v e. (UnivVtx ` G ) ) )
4	3	pm5.32i 669	. 2 |- ( ( G e. USGraph /\ G e. ComplGraph ) <-> ( G e. USGraph /\ A. v e. V v e. (UnivVtx ` G ) ) )
5	1, 4	bitri 264	1 |- ( G e. ComplUSGraph <-> ( G e. USGraph /\ A. v e. V v e. (UnivVtx ` G ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  Vtxcvtx 25874   USGraph cusgr 26044  UnivVtxcuvtxa 26225  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-3an 1039  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-cplgr 26231  df-cusgr 26232

This theorem is referenced by: (None)