Theorem nb3grapr2 23534

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 977	. . . . 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 4064	. . . . . . . . . . 11 |- { C , A } = { A , C }
4	3	eleq1i 2531	. . . . . . . . . 10 |- ( { C , A } e. ran E <-> { A , C } e. ran E )
5	4	biimpi 194	. . . . . . . . 9 |- ( { C , A } e. ran E -> { A , C } e. ran E )
6	5	pm4.71i 632	. . . . . . . 8 |- ( { C , A } e. ran E <-> ( { C , A } e. ran E /\ { A , C } e. ran E ) )
7	6	anbi2i 694	. . . . . . 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 649	. . . . . . 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 252	. . . . . 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 695	. . . . 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 649	. . . . 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 249	. . . 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 261	. . 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 4064	. . . . . . . . . 10 |- { A , B } = { B , A }
15	14	eleq1i 2531	. . . . . . . . 9 |- ( { A , B } e. ran E <-> { B , A } e. ran E )
16	15	biimpi 194	. . . . . . . 8 |- ( { A , B } e. ran E -> { B , A } e. ran E )
17	16	pm4.71i 632	. . . . . . 7 |- ( { A , B } e. ran E <-> ( { A , B } e. ran E /\ { B , A } e. ran E ) )
18	17	anbi1i 695	. . . . . 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 967	. . . . . 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 252	. . . . 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 4064	. . . . . . . . . 10 |- { B , C } = { C , B }
22	21	eleq1i 2531	. . . . . . . . 9 |- ( { B , C } e. ran E <-> { C , B } e. ran E )
23	22	biimpi 194	. . . . . . . 8 |- ( { B , C } e. ran E -> { C , B } e. ran E )
24	23	pm4.71i 632	. . . . . . 7 |- ( { B , C } e. ran E <-> ( { B , C } e. ran E /\ { C , B } e. ran E ) )
25	24	anbi2i 694	. . . . . 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 969	. . . . . 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 252	. . . . 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 697	. . . 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 1299	. . . 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 249	. . 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 261	. 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 23531	. . . . 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 201	. . . 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 972	. . . . . . 7 |- ( ( A e. X /\ B e. Y /\ C e. Z ) <-> ( B e. Y /\ A e. X /\ C e. Z ) )
35	34	biimpi 194	. . . . . 6 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( B e. Y /\ A e. X /\ C e. Z ) )
36		tpcoma 4082	. . . . . . . . 9 |- { A , B , C } = { B , A , C }
37	36	eqeq2i 2472	. . . . . . . 8 |- ( V = { A , B , C } <-> V = { B , A , C } )
38	37	biimpi 194	. . . . . . 7 |- ( V = { A , B , C } -> V = { B , A , C } )
39	38	anim1i 568	. . . . . 6 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V = { B , A , C } /\ V USGrph E ) )
40		nb3graprlem1 23531	. . . . . 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 477	. . . . 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 201	. . . 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 970	. . . . . . 7 |- ( ( C e. Z /\ A e. X /\ B e. Y ) <-> ( A e. X /\ B e. Y /\ C e. Z ) )
44	43	biimpri 206	. . . . . 6 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( C e. Z /\ A e. X /\ B e. Y ) )
45		tprot 4081	. . . . . . . . . 10 |- { C , A , B } = { A , B , C }
46	45	eqcomi 2467	. . . . . . . . 9 |- { A , B , C } = { C , A , B }
47	46	eqeq2i 2472	. . . . . . . 8 |- ( V = { A , B , C } <-> V = { C , A , B } )
48	47	biimpi 194	. . . . . . 7 |- ( V = { A , B , C } -> V = { C , A , B } )
49	48	anim1i 568	. . . . . 6 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V = { C , A , B } /\ V USGrph E ) )
50		nb3graprlem1 23531	. . . . . 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 477	. . . . 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 201	. . . 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 1290	. . 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 1008	. 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 253	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 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2648   {cpr 3990   {ctp 3992   <.cop 3994   class class class wbr 4403   ran crn 4952  (class class class)co 6203   USGrph cusg 23436   Neighbors cnbgra 23501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-hash 12224  df-usgra 23438  df-nbgra 23504

This theorem is referenced by: (None)