Theorem nb3grapr2 25182

Description: The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)

Assertion

Ref	Expression
nb3grapr2	|- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( { A , B } e. ran E /\ { B , C } e. ran E /\ { C , A } e. ran E ) <-> ( ( <. V , E >. Neighbors A ) = { B , C } /\ ( <. V , E >. Neighbors B ) = { A , C } /\ ( <. V , E >. Neighbors C ) = { A , B } ) ) )

Proof of Theorem nb3grapr2

Step	Hyp	Ref	Expression
1		3anan32 997	. . . . 5 |- ( ( { A , B } e. ran E /\ { B , C } e. ran E /\ { C , A } e. ran E ) <-> ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ { B , C } e. ran E ) )
2	1	a1i 11	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( { A , B } e. ran E /\ { B , C } e. ran E /\ { C , A } e. ran E ) <-> ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ { B , C } e. ran E ) ) )
3		prcom 4050	. . . . . . . . . . 11 |- { C , A } = { A , C }
4	3	eleq1i 2520	. . . . . . . . . 10 |- ( { C , A } e. ran E <-> { A , C } e. ran E )
5	4	biimpi 198	. . . . . . . . 9 |- ( { C , A } e. ran E -> { A , C } e. ran E )
6	5	pm4.71i 638	. . . . . . . 8 |- ( { C , A } e. ran E <-> ( { C , A } e. ran E /\ { A , C } e. ran E ) )
7	6	anbi2i 700	. . . . . . 7 |- ( ( { A , B } e. ran E /\ { C , A } e. ran E ) <-> ( { A , B } e. ran E /\ ( { C , A } e. ran E /\ { A , C } e. ran E ) ) )
8		anass 655	. . . . . . 7 |- ( ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ { A , C } e. ran E ) <-> ( { A , B } e. ran E /\ ( { C , A } e. ran E /\ { A , C } e. ran E ) ) )
9	7, 8	bitr4i 256	. . . . . 6 |- ( ( { A , B } e. ran E /\ { C , A } e. ran E ) <-> ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ { A , C } e. ran E ) )
10	9	anbi1i 701	. . . . 5 |- ( ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ { B , C } e. ran E ) <-> ( ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ { A , C } e. ran E ) /\ { B , C } e. ran E ) )
11		anass 655	. . . . 5 |- ( ( ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ { A , C } e. ran E ) /\ { B , C } e. ran E ) <-> ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ ( { A , C } e. ran E /\ { B , C } e. ran E ) ) )
12	10, 11	bitri 253	. . . 4 |- ( ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ { B , C } e. ran E ) <-> ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ ( { A , C } e. ran E /\ { B , C } e. ran E ) ) )
13	2, 12	syl6bb 265	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( { A , B } e. ran E /\ { B , C } e. ran E /\ { C , A } e. ran E ) <-> ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ ( { A , C } e. ran E /\ { B , C } e. ran E ) ) ) )
14		prcom 4050	. . . . . . . . . 10 |- { A , B } = { B , A }
15	14	eleq1i 2520	. . . . . . . . 9 |- ( { A , B } e. ran E <-> { B , A } e. ran E )
16	15	biimpi 198	. . . . . . . 8 |- ( { A , B } e. ran E -> { B , A } e. ran E )
17	16	pm4.71i 638	. . . . . . 7 |- ( { A , B } e. ran E <-> ( { A , B } e. ran E /\ { B , A } e. ran E ) )
18	17	anbi1i 701	. . . . . 6 |- ( ( { A , B } e. ran E /\ { C , A } e. ran E ) <-> ( ( { A , B } e. ran E /\ { B , A } e. ran E ) /\ { C , A } e. ran E ) )
19		df-3an 987	. . . . . 6 |- ( ( { A , B } e. ran E /\ { B , A } e. ran E /\ { C , A } e. ran E ) <-> ( ( { A , B } e. ran E /\ { B , A } e. ran E ) /\ { C , A } e. ran E ) )
20	18, 19	bitr4i 256	. . . . 5 |- ( ( { A , B } e. ran E /\ { C , A } e. ran E ) <-> ( { A , B } e. ran E /\ { B , A } e. ran E /\ { C , A } e. ran E ) )
21		prcom 4050	. . . . . . . . . 10 |- { B , C } = { C , B }
22	21	eleq1i 2520	. . . . . . . . 9 |- ( { B , C } e. ran E <-> { C , B } e. ran E )
23	22	biimpi 198	. . . . . . . 8 |- ( { B , C } e. ran E -> { C , B } e. ran E )
24	23	pm4.71i 638	. . . . . . 7 |- ( { B , C } e. ran E <-> ( { B , C } e. ran E /\ { C , B } e. ran E ) )
25	24	anbi2i 700	. . . . . 6 |- ( ( { A , C } e. ran E /\ { B , C } e. ran E ) <-> ( { A , C } e. ran E /\ ( { B , C } e. ran E /\ { C , B } e. ran E ) ) )
26		3anass 989	. . . . . 6 |- ( ( { A , C } e. ran E /\ { B , C } e. ran E /\ { C , B } e. ran E ) <-> ( { A , C } e. ran E /\ ( { B , C } e. ran E /\ { C , B } e. ran E ) ) )
27	25, 26	bitr4i 256	. . . . 5 |- ( ( { A , C } e. ran E /\ { B , C } e. ran E ) <-> ( { A , C } e. ran E /\ { B , C } e. ran E /\ { C , B } e. ran E ) )
28	20, 27	anbi12i 703	. . . 4 |- ( ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ ( { A , C } e. ran E /\ { B , C } e. ran E ) ) <-> ( ( { A , B } e. ran E /\ { B , A } e. ran E /\ { C , A } e. ran E ) /\ ( { A , C } e. ran E /\ { B , C } e. ran E /\ { C , B } e. ran E ) ) )
29		an6 1348	. . . 4 |- ( ( ( { A , B } e. ran E /\ { B , A } e. ran E /\ { C , A } e. ran E ) /\ ( { A , C } e. ran E /\ { B , C } e. ran E /\ { C , B } e. ran E ) ) <-> ( ( { A , B } e. ran E /\ { A , C } e. ran E ) /\ ( { B , A } e. ran E /\ { B , C } e. ran E ) /\ ( { C , A } e. ran E /\ { C , B } e. ran E ) ) )
30	28, 29	bitri 253	. . 3 |- ( ( ( { A , B } e. ran E /\ { C , A } e. ran E ) /\ ( { A , C } e. ran E /\ { B , C } e. ran E ) ) <-> ( ( { A , B } e. ran E /\ { A , C } e. ran E ) /\ ( { B , A } e. ran E /\ { B , C } e. ran E ) /\ ( { C , A } e. ran E /\ { C , B } e. ran E ) ) )
31	13, 30	syl6bb 265	. 2 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( { A , B } e. ran E /\ { B , C } e. ran E /\ { C , A } e. ran E ) <-> ( ( { A , B } e. ran E /\ { A , C } e. ran E ) /\ ( { B , A } e. ran E /\ { B , C } e. ran E ) /\ ( { C , A } e. ran E /\ { C , B } e. ran E ) ) ) )
32		nb3graprlem1 25179	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> ( ( <. V , E >. Neighbors A ) = { B , C } <-> ( { A , B } e. ran E /\ { A , C } e. ran E ) ) )
33	32	bicomd 205	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> ( ( { A , B } e. ran E /\ { A , C } e. ran E ) <-> ( <. V , E >. Neighbors A ) = { B , C } ) )
34		3ancoma 992	. . . . . . 7 |- ( ( A e. X /\ B e. Y /\ C e. Z ) <-> ( B e. Y /\ A e. X /\ C e. Z ) )
35	34	biimpi 198	. . . . . 6 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( B e. Y /\ A e. X /\ C e. Z ) )
36		tpcoma 4068	. . . . . . . . 9 |- { A , B , C } = { B , A , C }
37	36	eqeq2i 2463	. . . . . . . 8 |- ( V = { A , B , C } <-> V = { B , A , C } )
38	37	biimpi 198	. . . . . . 7 |- ( V = { A , B , C } -> V = { B , A , C } )
39	38	anim1i 572	. . . . . 6 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V = { B , A , C } /\ V USGrph E ) )
40		nb3graprlem1 25179	. . . . . 6 |- ( ( ( B e. Y /\ A e. X /\ C e. Z ) /\ ( V = { B , A , C } /\ V USGrph E ) ) -> ( ( <. V , E >. Neighbors B ) = { A , C } <-> ( { B , A } e. ran E /\ { B , C } e. ran E ) ) )
41	35, 39, 40	syl2an 480	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> ( ( <. V , E >. Neighbors B ) = { A , C } <-> ( { B , A } e. ran E /\ { B , C } e. ran E ) ) )
42	41	bicomd 205	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> ( ( { B , A } e. ran E /\ { B , C } e. ran E ) <-> ( <. V , E >. Neighbors B ) = { A , C } ) )
43		3anrot 990	. . . . . . 7 |- ( ( C e. Z /\ A e. X /\ B e. Y ) <-> ( A e. X /\ B e. Y /\ C e. Z ) )
44	43	biimpri 210	. . . . . 6 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( C e. Z /\ A e. X /\ B e. Y ) )
45		tprot 4067	. . . . . . . . . 10 |- { C , A , B } = { A , B , C }
46	45	eqcomi 2460	. . . . . . . . 9 |- { A , B , C } = { C , A , B }
47	46	eqeq2i 2463	. . . . . . . 8 |- ( V = { A , B , C } <-> V = { C , A , B } )
48	47	biimpi 198	. . . . . . 7 |- ( V = { A , B , C } -> V = { C , A , B } )
49	48	anim1i 572	. . . . . 6 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V = { C , A , B } /\ V USGrph E ) )
50		nb3graprlem1 25179	. . . . . 6 |- ( ( ( C e. Z /\ A e. X /\ B e. Y ) /\ ( V = { C , A , B } /\ V USGrph E ) ) -> ( ( <. V , E >. Neighbors C ) = { A , B } <-> ( { C , A } e. ran E /\ { C , B } e. ran E ) ) )
51	44, 49, 50	syl2an 480	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> ( ( <. V , E >. Neighbors C ) = { A , B } <-> ( { C , A } e. ran E /\ { C , B } e. ran E ) ) )
52	51	bicomd 205	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> ( ( { C , A } e. ran E /\ { C , B } e. ran E ) <-> ( <. V , E >. Neighbors C ) = { A , B } ) )
53	33, 42, 52	3anbi123d 1339	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> ( ( ( { A , B } e. ran E /\ { A , C } e. ran E ) /\ ( { B , A } e. ran E /\ { B , C } e. ran E ) /\ ( { C , A } e. ran E /\ { C , B } e. ran E ) ) <-> ( ( <. V , E >. Neighbors A ) = { B , C } /\ ( <. V , E >. Neighbors B ) = { A , C } /\ ( <. V , E >. Neighbors C ) = { A , B } ) ) )
54	53	3adant3 1028	. 2 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( { A , B } e. ran E /\ { A , C } e. ran E ) /\ ( { B , A } e. ran E /\ { B , C } e. ran E ) /\ ( { C , A } e. ran E /\ { C , B } e. ran E ) ) <-> ( ( <. V , E >. Neighbors A ) = { B , C } /\ ( <. V , E >. Neighbors B ) = { A , C } /\ ( <. V , E >. Neighbors C ) = { A , B } ) ) )
55	31, 54	bitrd 257	1 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( { A , B } e. ran E /\ { B , C } e. ran E /\ { C , A } e. ran E ) <-> ( ( <. V , E >. Neighbors A ) = { B , C } /\ ( <. V , E >. Neighbors B ) = { A , C } /\ ( <. V , E >. Neighbors C ) = { A , B } ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   {cpr 3970   {ctp 3972   <.cop 3974   class class class wbr 4402   ran crn 4835  (class class class)co 6290   USGrph cusg 25057   Neighbors cnbgra 25145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-hash 12516  df-usgra 25060  df-nbgra 25148

This theorem is referenced by: (None)