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Theorem nb3grapr 25173
 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
nb3grapr USGrph Neighbors
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)

Proof of Theorem nb3grapr
StepHypRef Expression
1 id 23 . . . . . 6
2 prcom 4076 . . . . . . . . . 10
32eleq1i 2500 . . . . . . . . 9
4 prcom 4076 . . . . . . . . . 10
54eleq1i 2500 . . . . . . . . 9
6 prcom 4076 . . . . . . . . . 10
76eleq1i 2500 . . . . . . . . 9
83, 5, 73anbi123i 1195 . . . . . . . 8
9 3anrot 988 . . . . . . . 8
108, 9bitr4i 256 . . . . . . 7
1110a1i 11 . . . . . 6
121, 11biadan2 647 . . . . 5
13 an6 1345 . . . . 5
1412, 13bitri 253 . . . 4
1514a1i 11 . . 3 USGrph
16 nb3graprlem1 25171 . . . . 5 USGrph Neighbors
17 3anrot 988 . . . . . . 7
1817biimpi 198 . . . . . 6
19 tprot 4093 . . . . . . . . 9
2019eqeq2i 2441 . . . . . . . 8
2120biimpi 198 . . . . . . 7
2221anim1i 571 . . . . . 6 USGrph USGrph
23 nb3graprlem1 25171 . . . . . 6 USGrph Neighbors
2418, 22, 23syl2an 480 . . . . 5 USGrph Neighbors
25 3anrot 988 . . . . . . 7
2625biimpri 210 . . . . . 6
27 tprot 4093 . . . . . . . . . 10
2827eqcomi 2436 . . . . . . . . 9
2928eqeq2i 2441 . . . . . . . 8
3029biimpi 198 . . . . . . 7
3130anim1i 571 . . . . . 6 USGrph USGrph
32 nb3graprlem1 25171 . . . . . 6 USGrph Neighbors
3326, 31, 32syl2an 480 . . . . 5 USGrph Neighbors
3416, 24, 333anbi123d 1336 . . . 4 USGrph Neighbors Neighbors Neighbors
35343adant3 1026 . . 3 USGrph Neighbors Neighbors Neighbors
36 nb3graprlem2 25172 . . . 4 USGrph Neighbors Neighbors
3720anbi1i 700 . . . . 5 USGrph USGrph
38 necom 2694 . . . . . . 7
39 necom 2694 . . . . . . 7
40 biid 240 . . . . . . 7
4138, 39, 403anbi123i 1195 . . . . . 6
42 3anrot 988 . . . . . 6
4341, 42bitr4i 256 . . . . 5
44 nb3graprlem2 25172 . . . . 5 USGrph Neighbors Neighbors
4517, 37, 43, 44syl3anb 1308 . . . 4 USGrph Neighbors Neighbors
46 id 23 . . . . . . 7
4746, 28syl6eq 2480 . . . . . 6
4847anim1i 571 . . . . 5 USGrph USGrph
49 3anrot 988 . . . . . . 7
50 necom 2694 . . . . . . . 8
51 biid 240 . . . . . . . 8
5239, 50, 513anbi123i 1195 . . . . . . 7
5349, 52bitri 253 . . . . . 6
5453biimpi 198 . . . . 5
55 nb3graprlem2 25172 . . . . 5 USGrph Neighbors Neighbors
5626, 48, 54, 55syl3an 1307 . . . 4 USGrph Neighbors Neighbors
5736, 45, 563anbi123d 1336 . . 3 USGrph Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
5815, 35, 573bitr2d 285 . 2 USGrph Neighbors Neighbors Neighbors
59 oveq2 6311 . . . . . 6 Neighbors Neighbors
6059eqeq1d 2425 . . . . 5 Neighbors Neighbors
61602rexbidv 2947 . . . 4 Neighbors Neighbors
62 oveq2 6311 . . . . . 6 Neighbors Neighbors
6362eqeq1d 2425 . . . . 5 Neighbors Neighbors
64632rexbidv 2947 . . . 4 Neighbors Neighbors
65 oveq2 6311 . . . . . 6 Neighbors Neighbors
6665eqeq1d 2425 . . . . 5 Neighbors Neighbors
67662rexbidv 2947 . . . 4 Neighbors Neighbors
6861, 64, 67raltpg 4049 . . 3 Neighbors Neighbors Neighbors Neighbors
69683ad2ant1 1027 . 2 USGrph Neighbors Neighbors Neighbors Neighbors
70 raleq 3026 . . . . 5 Neighbors Neighbors
7170bicomd 205 . . . 4 Neighbors Neighbors
7271adantr 467 . . 3 USGrph Neighbors Neighbors
73723ad2ant2 1028 . 2 USGrph Neighbors Neighbors
7458, 69, 733bitr2d 285 1 USGrph Neighbors
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wa 371   w3a 983   wceq 1438   wcel 1869   wne 2619  wral 2776  wrex 2777   cdif 3434  csn 3997  cpr 3999  ctp 4001  cop 4003   class class class wbr 4421   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:  cusgra3vnbpr  25185
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