Theorem iscplgredg 26313

Description: A graph is complete iff all vertices are connected with each other by (at least) one edge. (Contributed by AV, 10-Nov-2020.)

Hypotheses

Ref	Expression
iscplgr.v	|- V = (Vtx ` G )
iscplgredg.v	|- E = (Edg ` G )

Assertion

Ref	Expression
iscplgredg	|- ( G e. W -> ( G e. ComplGraph <-> A. v e. V A. n e. ( V \ { v } ) E. e e. E { v , n } C_ e ) )

Distinct variable groups:   v, G   v, V   n, G, v   n, V   v, W   e, E   e, G   e, V   e, W, n, v

Allowed substitution hints:   E( v, n)

Proof of Theorem iscplgredg

Step	Hyp	Ref	Expression
1		iscplgr.v	. . 3 |- V = (Vtx ` G )
2	1	iscplgrnb 26312	. 2 |- ( G e. W -> ( G e. ComplGraph <-> A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) ) )
3		df-3an 1039	. . . . . 6 |- ( ( ( n e. V /\ v e. V ) /\ n =/= v /\ E. e e. E { v , n } C_ e ) <-> ( ( ( n e. V /\ v e. V ) /\ n =/= v ) /\ E. e e. E { v , n } C_ e ) )
4	3	a1i 11	. . . . 5 |- ( ( ( G e. W /\ v e. V ) /\ n e. ( V \ { v } ) ) -> ( ( ( n e. V /\ v e. V ) /\ n =/= v /\ E. e e. E { v , n } C_ e ) <-> ( ( ( n e. V /\ v e. V ) /\ n =/= v ) /\ E. e e. E { v , n } C_ e ) ) )
5		iscplgredg.v	. . . . . . 7 |- E = (Edg ` G )
6	1, 5	nbgrel 26238	. . . . . 6 |- ( G e. W -> ( n e. ( G NeighbVtx v ) <-> ( ( n e. V /\ v e. V ) /\ n =/= v /\ E. e e. E { v , n } C_ e ) ) )
7	6	ad2antrr 762	. . . . 5 |- ( ( ( G e. W /\ v e. V ) /\ n e. ( V \ { v } ) ) -> ( n e. ( G NeighbVtx v ) <-> ( ( n e. V /\ v e. V ) /\ n =/= v /\ E. e e. E { v , n } C_ e ) ) )
8		eldifsn 4317	. . . . . . 7 |- ( n e. ( V \ { v } ) <-> ( n e. V /\ n =/= v ) )
9		simpr 477	. . . . . . . . 9 |- ( ( G e. W /\ v e. V ) -> v e. V )
10		simpl 473	. . . . . . . . 9 |- ( ( n e. V /\ n =/= v ) -> n e. V )
11	9, 10	anim12ci 591	. . . . . . . 8 |- ( ( ( G e. W /\ v e. V ) /\ ( n e. V /\ n =/= v ) ) -> ( n e. V /\ v e. V ) )
12		simprr 796	. . . . . . . 8 |- ( ( ( G e. W /\ v e. V ) /\ ( n e. V /\ n =/= v ) ) -> n =/= v )
13	11, 12	jca 554	. . . . . . 7 |- ( ( ( G e. W /\ v e. V ) /\ ( n e. V /\ n =/= v ) ) -> ( ( n e. V /\ v e. V ) /\ n =/= v ) )
14	8, 13	sylan2b 492	. . . . . 6 |- ( ( ( G e. W /\ v e. V ) /\ n e. ( V \ { v } ) ) -> ( ( n e. V /\ v e. V ) /\ n =/= v ) )
15	14	biantrurd 529	. . . . 5 |- ( ( ( G e. W /\ v e. V ) /\ n e. ( V \ { v } ) ) -> ( E. e e. E { v , n } C_ e <-> ( ( ( n e. V /\ v e. V ) /\ n =/= v ) /\ E. e e. E { v , n } C_ e ) ) )
16	4, 7, 15	3bitr4d 300	. . . 4 |- ( ( ( G e. W /\ v e. V ) /\ n e. ( V \ { v } ) ) -> ( n e. ( G NeighbVtx v ) <-> E. e e. E { v , n } C_ e ) )
17	16	ralbidva 2985	. . 3 |- ( ( G e. W /\ v e. V ) -> ( A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) <-> A. n e. ( V \ { v } ) E. e e. E { v , n } C_ e ) )
18	17	ralbidva 2985	. 2 |- ( G e. W -> ( A. v e. V A. n e. ( V \ { v } ) n e. ( G NeighbVtx v ) <-> A. v e. V A. n e. ( V \ { v } ) E. e e. E { v , n } C_ e ) )
19	2, 18	bitrd 268	1 |- ( G e. W -> ( G e. ComplGraph <-> A. v e. V A. n e. ( V \ { v } ) E. e e. E { v , n } C_ e ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   NeighbVtx cnbgr 26224  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-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  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-nbgr 26228  df-uvtxa 26230  df-cplgr 26231

This theorem is referenced by:  cplgrop  26333  cusconngr  27051