Theorem cusgra2v 25269

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 25144	. . . . 5 |- ( { A , B } USGrph E -> ( { A , B } e. _V /\ E e. _V ) )
2	1	adantr 472	. . . 4 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( { A , B } e. _V /\ E e. _V ) )
3		iscusgra 25263	. . . 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 17	. . 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 1027	. . . . . 6 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( A e. V /\ B e. W ) )
6	5	adantl 473	. . . . 5 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( A e. V /\ B e. W ) )
7		sneq 3969	. . . . . . . 8 |- ( k = A -> { k } = { A } )
8	7	difeq2d 3540	. . . . . . 7 |- ( k = A -> ( { A , B } \ { k } ) = ( { A , B } \ { A } ) )
9		preq2 4043	. . . . . . . 8 |- ( k = A -> { n , k } = { n , A } )
10	9	eleq1d 2533	. . . . . . 7 |- ( k = A -> ( { n , k } e. ran E <-> { n , A } e. ran E ) )
11	8, 10	raleqbidv 2987	. . . . . 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 3969	. . . . . . . 8 |- ( k = B -> { k } = { B } )
13	12	difeq2d 3540	. . . . . . 7 |- ( k = B -> ( { A , B } \ { k } ) = ( { A , B } \ { B } ) )
14		preq2 4043	. . . . . . . 8 |- ( k = B -> { n , k } = { n , B } )
15	14	eleq1d 2533	. . . . . . 7 |- ( k = B -> ( { n , k } e. ran E <-> { n , B } e. ran E ) )
16	13, 15	raleqbidv 2987	. . . . . 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 4012	. . . . 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 17	. . . 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 512	. . . . 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 472	. . . 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 4099	. . . . . . . . . 10 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )
22	21	3ad2ant3 1053	. . . . . . . . 9 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } \ { A } ) = { B } )
23	22	adantl 473	. . . . . . . 8 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( { A , B } \ { A } ) = { B } )
24	23	raleqdv 2979	. . . . . . 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 4042	. . . . . . . . . . 11 |- ( n = B -> { n , A } = { B , A } )
26	25	eleq1d 2533	. . . . . . . . . 10 |- ( n = B -> ( { n , A } e. ran E <-> { B , A } e. ran E ) )
27	26	ralsng 3997	. . . . . . . . 9 |- ( B e. W -> ( A. n e. { B } { n , A } e. ran E <-> { B , A } e. ran E ) )
28	27	3ad2ant2 1052	. . . . . . . 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 473	. . . . . . 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 261	. . . . . 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 4100	. . . . . . . . . 10 |- ( A =/= B -> ( { A , B } \ { B } ) = { A } )
32	31	3ad2ant3 1053	. . . . . . . . 9 |- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } \ { B } ) = { A } )
33	32	adantl 473	. . . . . . . 8 |- ( ( { A , B } USGrph E /\ ( A e. V /\ B e. W /\ A =/= B ) ) -> ( { A , B } \ { B } ) = { A } )
34	33	raleqdv 2979	. . . . . . 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 4042	. . . . . . . . . . 11 |- ( n = A -> { n , B } = { A , B } )
36	35	eleq1d 2533	. . . . . . . . . 10 |- ( n = A -> ( { n , B } e. ran E <-> { A , B } e. ran E ) )
37	36	ralsng 3997	. . . . . . . . 9 |- ( A e. V -> ( A. n e. { A } { n , B } e. ran E <-> { A , B } e. ran E ) )
38	37	3ad2ant1 1051	. . . . . . . 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 473	. . . . . . 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 261	. . . . . 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 725	. . . . 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 4041	. . . . . . . 8 |- { B , A } = { A , B }
43	42	eleq1i 2540	. . . . . . 7 |- ( { B , A } e. ran E <-> { A , B } e. ran E )
44	43	anbi1i 709	. . . . . 6 |- ( ( { B , A } e. ran E /\ { A , B } e. ran E ) <-> ( { A , B } e. ran E /\ { A , B } e. ran E ) )
45		anidm 656	. . . . . 6 |- ( ( { A , B } e. ran E /\ { A , B } e. ran E ) <-> { A , B } e. ran E )
46	44, 45	bitri 257	. . . . 5 |- ( ( { B , A } e. ran E /\ { A , B } e. ran E ) <-> { A , B } e. ran E )
47	41, 46	syl6bb 269	. . . 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 291	. . 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 261	. 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 442	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   _Vcvv 3031   \ cdif 3387   {csn 3959   {cpr 3961   class class class wbr 4395   ran crn 4840   USGrph cusg 25136   ComplUSGrph ccusgra 25225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-usgra 25139  df-cusgra 25228

This theorem is referenced by: (None)