Theorem cusgra2v 24126

Description: A graph with two (different) vertices is complete if and only if there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)

Assertion

Ref	Expression
cusgra2v	|- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } USGrph E -> ( { A , B } ComplUSGrph E <-> { A , B } e. ran E ) ) )

Proof of Theorem cusgra2v

Dummy variables k n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrav 24003	. . . . 5 |- ( { A , B } USGrph E -> ( { A , B } e. _V /\ E e. _V ) )
2	1	adantr 465	. . . 4 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( { A , B } e. _V /\ E e. _V ) )
3		iscusgra 24120	. . . 4 |- ( ( { A , B } e. _V /\ E e. _V ) -> ( { A , B } ComplUSGrph E <-> ( { A , B } USGrph E /\ A. k e. { A , B } A. n e. ( { A , B } \ { k } ) { n , k } e. ran E ) ) )
4	2, 3	syl 16	. . 3 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( { A , B } ComplUSGrph E <-> ( { A , B } USGrph E /\ A. k e. { A , B } A. n e. ( { A , B } \ { k } ) { n , k } e. ran E ) ) )
5		3simpa 988	. . . . . 6 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( A e. V /\ B e. W ) )
6	5	adantl 466	. . . . 5 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A e. V /\ B e. W ) )
7		sneq 4032	. . . . . . . 8 |- ( k = A -> { k } = { A } )
8	7	difeq2d 3617	. . . . . . 7 |- ( k = A -> ( { A , B } \ { k } ) = ( { A , B } \ { A } ) )
9		preq2 4102	. . . . . . . 8 |- ( k = A -> { n , k } = { n , A } )
10	9	eleq1d 2531	. . . . . . 7 |- ( k = A -> ( { n , k } e. ran E <-> { n , A } e. ran E ) )
11	8, 10	raleqbidv 3067	. . . . . 6 |- ( k = A -> ( A. n e. ( { A , B } \ { k } ) { n , k } e. ran E <-> A. n e. ( { A , B } \ { A } ) { n , A } e. ran E ) )
12		sneq 4032	. . . . . . . 8 |- ( k = B -> { k } = { B } )
13	12	difeq2d 3617	. . . . . . 7 |- ( k = B -> ( { A , B } \ { k } ) = ( { A , B } \ { B } ) )
14		preq2 4102	. . . . . . . 8 |- ( k = B -> { n , k } = { n , B } )
15	14	eleq1d 2531	. . . . . . 7 |- ( k = B -> ( { n , k } e. ran E <-> { n , B } e. ran E ) )
16	13, 15	raleqbidv 3067	. . . . . 6 |- ( k = B -> ( A. n e. ( { A , B } \ { k } ) { n , k } e. ran E <-> A. n e. ( { A , B } \ { B } ) { n , B } e. ran E ) )
17	11, 16	ralprg 4071	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( A. k e. { A , B } A. n e. ( { A , B } \ { k } ) { n , k } e. ran E <-> ( A. n e. ( { A , B } \ { A } ) { n , A } e. ran E /\ A. n e. ( { A , B } \ { B } ) { n , B } e. ran E ) ) )
18	6, 17	syl 16	. . . 4 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A. k e. { A , B } A. n e. ( { A , B } \ { k } ) { n , k } e. ran E <-> ( A. n e. ( { A , B } \ { A } ) { n , A } e. ran E /\ A. n e. ( { A , B } \ { B } ) { n , B } e. ran E ) ) )
19		ibar 504	. . . . 5 |- ( { A , B } USGrph E -> ( A. k e. { A , B } A. n e. ( { A , B } \ { k } ) { n , k } e. ran E <-> ( { A , B } USGrph E /\ A. k e. { A , B } A. n e. ( { A , B } \ { k } ) { n , k } e. ran E ) ) )
20	19	adantr 465	. . . 4 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A. k e. { A , B } A. n e. ( { A , B } \ { k } ) { n , k } e. ran E <-> ( { A , B } USGrph E /\ A. k e. { A , B } A. n e. ( { A , B } \ { k } ) { n , k } e. ran E ) ) )
21		difprsn1 4158	. . . . . . . . . 10 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )
22	21	3ad2ant3 1014	. . . . . . . . 9 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } \ { A } ) = { B } )
23	22	adantl 466	. . . . . . . 8 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( { A , B } \ { A } ) = { B } )
24	23	raleqdv 3059	. . . . . . 7 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A. n e. ( { A , B } \ { A } ) { n , A } e. ran E <-> A. n e. { B } { n , A } e. ran E ) )
25		preq1 4101	. . . . . . . . . . 11 |- ( n = B -> { n , A } = { B , A } )
26	25	eleq1d 2531	. . . . . . . . . 10 |- ( n = B -> ( { n , A } e. ran E <-> { B , A } e. ran E ) )
27	26	ralsng 4057	. . . . . . . . 9 |- ( B e. W -> ( A. n e. { B } { n , A } e. ran E <-> { B , A } e. ran E ) )
28	27	3ad2ant2 1013	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( A. n e. { B } { n , A } e. ran E <-> { B , A } e. ran E ) )
29	28	adantl 466	. . . . . . 7 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A. n e. { B } { n , A } e. ran E <-> { B , A } e. ran E ) )
30	24, 29	bitrd 253	. . . . . 6 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A. n e. ( { A , B } \ { A } ) { n , A } e. ran E <-> { B , A } e. ran E ) )
31		difprsn2 4159	. . . . . . . . . 10 |- ( A =/= B -> ( { A , B } \ { B } ) = { A } )
32	31	3ad2ant3 1014	. . . . . . . . 9 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } \ { B } ) = { A } )
33	32	adantl 466	. . . . . . . 8 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( { A , B } \ { B } ) = { A } )
34	33	raleqdv 3059	. . . . . . 7 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A. n e. ( { A , B } \ { B } ) { n , B } e. ran E <-> A. n e. { A } { n , B } e. ran E ) )
35		preq1 4101	. . . . . . . . . . 11 |- ( n = A -> { n , B } = { A , B } )
36	35	eleq1d 2531	. . . . . . . . . 10 |- ( n = A -> ( { n , B } e. ran E <-> { A , B } e. ran E ) )
37	36	ralsng 4057	. . . . . . . . 9 |- ( A e. V -> ( A. n e. { A } { n , B } e. ran E <-> { A , B } e. ran E ) )
38	37	3ad2ant1 1012	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( A. n e. { A } { n , B } e. ran E <-> { A , B } e. ran E ) )
39	38	adantl 466	. . . . . . 7 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A. n e. { A } { n , B } e. ran E <-> { A , B } e. ran E ) )
40	34, 39	bitrd 253	. . . . . 6 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A. n e. ( { A , B } \ { B } ) { n , B } e. ran E <-> { A , B } e. ran E ) )
41	30, 40	anbi12d 710	. . . . 5 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( ( A. n e. ( { A , B } \ { A } ) { n , A } e. ran E /\ A. n e. ( { A , B } \ { B } ) { n , B } e. ran E ) <-> ( { B , A } e. ran E /\ { A , B } e. ran E ) ) )
42		prcom 4100	. . . . . . . 8 |- { B , A } = { A , B }
43	42	eleq1i 2539	. . . . . . 7 |- ( { B , A } e. ran E <-> { A , B } e. ran E )
44	43	anbi1i 695	. . . . . 6 |- ( ( { B , A } e. ran E /\ { A , B } e. ran E ) <-> ( { A , B } e. ran E /\ { A , B } e. ran E ) )
45		anidm 644	. . . . . 6 |- ( ( { A , B } e. ran E /\ { A , B } e. ran E ) <-> { A , B } e. ran E )
46	44, 45	bitri 249	. . . . 5 |- ( ( { B , A } e. ran E /\ { A , B } e. ran E ) <-> { A , B } e. ran E )
47	41, 46	syl6bb 261	. . . 4 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( ( A. n e. ( { A , B } \ { A } ) { n , A } e. ran E /\ A. n e. ( { A , B } \ { B } ) { n , B } e. ran E ) <-> { A , B } e. ran E ) )
48	18, 20, 47	3bitr3d 283	. . 3 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( ( { A , B } USGrph E /\ A. k e. { A , B } A. n e. ( { A , B } \ { k } ) { n , k } e. ran E ) <-> { A , B } e. ran E ) )
49	4, 48	bitrd 253	. 2 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( { A , B } ComplUSGrph E <-> { A , B } e. ran E ) )
50	49	expcom 435	1 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } USGrph E -> ( { A , B } ComplUSGrph E <-> { A , B } e. ran E ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   =/= wne 2657   A.wral 2809   _Vcvv 3108   \ cdif 3468   {csn 4022   {cpr 4024   class class class wbr 4442   ran crn 4995   USGrph cusg 23995   ComplUSGrph ccusgra 24082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-dm 5004  df-rn 5005  df-usgra 23998  df-cusgra 24085

This theorem is referenced by: (None)