Theorem cplgr0v 26323

Description: A null graph (with no vertices) is a complete graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)

Hypothesis

Ref	Expression
cplgr0v.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
cplgr0v	|- ( ( G e. W /\ V = (/) ) -> G e. ComplGraph )

Proof of Theorem cplgr0v

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rzal 4073	. . 3 |- ( V = (/) -> A. v e. V v e. (UnivVtx ` G ) )
2	1	adantl 482	. 2 |- ( ( G e. W /\ V = (/) ) -> A. v e. V v e. (UnivVtx ` G ) )
3		cplgr0v.v	. . . 4 |- V = (Vtx ` G )
4	3	iscplgr 26310	. . 3 |- ( G e. W -> ( G e. ComplGraph <-> A. v e. V v e. (UnivVtx ` G ) ) )
5	4	adantr 481	. 2 |- ( ( G e. W /\ V = (/) ) -> ( G e. ComplGraph <-> A. v e. V v e. (UnivVtx ` G ) ) )
6	2, 5	mpbird 247	1 |- ( ( G e. W /\ V = (/) ) -> G e. ComplGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   ` cfv 5888  Vtxcvtx 25874  UnivVtxcuvtxa 26225  ComplGraphccplgr 26226

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-ne 2795  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

This theorem is referenced by:  cusgr0v  26324