Theorem nb3grapr2 25174

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 995	. . . . 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 4076	. . . . . . . . . . 11 |- { C , A } = { A , C }
4	3	eleq1i 2500	. . . . . . . . . 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 637	. . . . . . . 8 |- ( { C , A } e. ran E <-> ( { C , A } e. ran E /\ { A , C } e. ran E ) )
7	6	anbi2i 699	. . . . . . 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 654	. . . . . . 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 700	. . . . 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 654	. . . . 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 4076	. . . . . . . . . 10 |- { A , B } = { B , A }
15	14	eleq1i 2500	. . . . . . . . 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 637	. . . . . . 7 |- ( { A , B } e. ran E <-> ( { A , B } e. ran E /\ { B , A } e. ran E ) )
18	17	anbi1i 700	. . . . . 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 985	. . . . . 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 4076	. . . . . . . . . 10 |- { B , C } = { C , B }
22	21	eleq1i 2500	. . . . . . . . 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 637	. . . . . . 7 |- ( { B , C } e. ran E <-> ( { B , C } e. ran E /\ { C , B } e. ran E ) )
25	24	anbi2i 699	. . . . . 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 987	. . . . . 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 702	. . . 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 1345	. . . 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 25171	. . . . 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 990	. . . . . . 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 4094	. . . . . . . . 9 |- { A , B , C } = { B , A , C }
37	36	eqeq2i 2441	. . . . . . . 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 571	. . . . . 6 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V = { B , A , C } /\ V USGrph E ) )
40		nb3graprlem1 25171	. . . . . 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 988	. . . . . . 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 4093	. . . . . . . . . 10 |- { C , A , B } = { A , B , C }
46	45	eqcomi 2436	. . . . . . . . 9 |- { A , B , C } = { C , A , B }
47	46	eqeq2i 2441	. . . . . . . 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 571	. . . . . 6 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V = { C , A , B } /\ V USGrph E ) )
50		nb3graprlem1 25171	. . . . . 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 1336	. . 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 1026	. 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 983   = wceq 1438   e. wcel 1869   =/= wne 2619   {cpr 3999   {ctp 4001   <.cop 4003   class class class wbr 4421   ran crn 4852  (class class class)co 6303   USGrph cusg 25049   Neighbors cnbgra 25137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-hash 12517  df-usgra 25052  df-nbgra 25140

This theorem is referenced by: (None)