Theorem nb3grapr2 24600

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 983	. . . . 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 4039	. . . . . . . . . . 11 |- { C , A } = { A , C }
4	3	eleq1i 2473	. . . . . . . . . 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 630	. . . . . . . 8 |- ( { C , A } e. ran E <-> ( { C , A } e. ran E /\ { A , C } e. ran E ) )
7	6	anbi2i 692	. . . . . . 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 647	. . . . . . 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 693	. . . . 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 647	. . . . 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 4039	. . . . . . . . . 10 |- { A , B } = { B , A }
15	14	eleq1i 2473	. . . . . . . . 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 630	. . . . . . 7 |- ( { A , B } e. ran E <-> ( { A , B } e. ran E /\ { B , A } e. ran E ) )
18	17	anbi1i 693	. . . . . 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 973	. . . . . 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 4039	. . . . . . . . . 10 |- { B , C } = { C , B }
22	21	eleq1i 2473	. . . . . . . . 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 630	. . . . . . 7 |- ( { B , C } e. ran E <-> ( { B , C } e. ran E /\ { C , B } e. ran E ) )
25	24	anbi2i 692	. . . . . 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 975	. . . . . 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 695	. . . 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 1306	. . . 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 24597	. . . . 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 978	. . . . . . 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 4057	. . . . . . . . 9 |- { A , B , C } = { B , A , C }
37	36	eqeq2i 2414	. . . . . . . 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 566	. . . . . 6 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V = { B , A , C } /\ V USGrph E ) )
40		nb3graprlem1 24597	. . . . . 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 475	. . . . 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 976	. . . . . . 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 4056	. . . . . . . . . 10 |- { C , A , B } = { A , B , C }
46	45	eqcomi 2409	. . . . . . . . 9 |- { A , B , C } = { C , A , B }
47	46	eqeq2i 2414	. . . . . . . 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 566	. . . . . 6 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V = { C , A , B } /\ V USGrph E ) )
50		nb3graprlem1 24597	. . . . . 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 475	. . . . 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 1297	. . 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 1014	. 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 367   /\ w3a 971   = wceq 1399   e. wcel 1836   =/= wne 2591   {cpr 3963   {ctp 3965   <.cop 3967   class class class wbr 4384   ran crn 4931  (class class class)co 6218   USGrph cusg 24476   Neighbors cnbgra 24563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-1o 7070  df-oadd 7074  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-card 8255  df-cda 8483  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-nn 10475  df-2 10533  df-n0 10735  df-z 10804  df-uz 11024  df-fz 11616  df-hash 12331  df-usgra 24479  df-nbgra 24566

This theorem is referenced by: (None)