Theorem cusgruvtxb 26318

Description: A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) (Revised by AV, 1-Nov-2020.)

Hypothesis

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

Assertion

Ref	Expression
cusgruvtxb	|- ( G e. USGraph -> ( G e. ComplUSGraph <-> (UnivVtx ` G ) = V ) )

Proof of Theorem cusgruvtxb

Step	Hyp	Ref	Expression
1		iscusgr 26314	. 2 |- ( G e. ComplUSGraph <-> ( G e. USGraph /\ G e. ComplGraph ) )
2		ibar 525	. . 3 |- ( G e. USGraph -> ( G e. ComplGraph <-> ( G e. USGraph /\ G e. ComplGraph ) ) )
3		iscusgrvtx.v	. . . 4 |- V = (Vtx ` G )
4	3	cplgruvtxb 26311	. . 3 |- ( G e. USGraph -> ( G e. ComplGraph <-> (UnivVtx ` G ) = V ) )
5	2, 4	bitr3d 270	. 2 |- ( G e. USGraph -> ( ( G e. USGraph /\ G e. ComplGraph ) <-> (UnivVtx ` G ) = V ) )
6	1, 5	syl5bb 272	1 |- ( G e. USGraph -> ( G e. ComplUSGraph <-> (UnivVtx ` G ) = V ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` 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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-uvtxa 26230  df-cplgr 26231  df-cusgr 26232

This theorem is referenced by:  vdiscusgrb  26426  vdiscusgr  26427