Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nb3grapr Structured version   Unicode version

Theorem nb3grapr 21462
 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 20 . . . . . 6
2 prcom 3882 . . . . . . . . . 10
32eleq1i 2499 . . . . . . . . 9
4 prcom 3882 . . . . . . . . . 10
54eleq1i 2499 . . . . . . . . 9
6 prcom 3882 . . . . . . . . . 10
76eleq1i 2499 . . . . . . . . 9
83, 5, 73anbi123i 1142 . . . . . . . 8
9 3anrot 941 . . . . . . . 8
108, 9bitr4i 244 . . . . . . 7
1110a1i 11 . . . . . 6
121, 11biadan2 624 . . . . 5
13 an6 1263 . . . . 5
1412, 13bitri 241 . . . 4
1514a1i 11 . . 3 USGrph
16 nb3graprlem1 21460 . . . . 5 USGrph Neighbors
17 3anrot 941 . . . . . . 7
1817biimpi 187 . . . . . 6
19 tprot 3899 . . . . . . . . 9
2019eqeq2i 2446 . . . . . . . 8
2120biimpi 187 . . . . . . 7
2221anim1i 552 . . . . . 6 USGrph USGrph
23 nb3graprlem1 21460 . . . . . 6 USGrph Neighbors
2418, 22, 23syl2an 464 . . . . 5 USGrph Neighbors
25 3anrot 941 . . . . . . 7
2625biimpri 198 . . . . . 6
27 tprot 3899 . . . . . . . . . 10
2827eqcomi 2440 . . . . . . . . 9
2928eqeq2i 2446 . . . . . . . 8
3029biimpi 187 . . . . . . 7
3130anim1i 552 . . . . . 6 USGrph USGrph
32 nb3graprlem1 21460 . . . . . 6 USGrph Neighbors
3326, 31, 32syl2an 464 . . . . 5 USGrph Neighbors
3416, 24, 333anbi123d 1254 . . . 4 USGrph Neighbors Neighbors Neighbors
35343adant3 977 . . 3 USGrph Neighbors Neighbors Neighbors
36 nb3graprlem2 21461 . . . 4 USGrph Neighbors Neighbors
3720anbi1i 677 . . . . 5 USGrph USGrph
38 necom 2685 . . . . . . 7
39 necom 2685 . . . . . . 7
40 biid 228 . . . . . . 7
4138, 39, 403anbi123i 1142 . . . . . 6
42 3anrot 941 . . . . . 6
4341, 42bitr4i 244 . . . . 5
44 nb3graprlem2 21461 . . . . 5 USGrph Neighbors Neighbors
4517, 37, 43, 44syl3anb 1227 . . . 4 USGrph Neighbors Neighbors
46 id 20 . . . . . . 7
4746, 28syl6eq 2484 . . . . . 6
4847anim1i 552 . . . . 5 USGrph USGrph
49 3anrot 941 . . . . . . 7
50 necom 2685 . . . . . . . 8
51 biid 228 . . . . . . . 8
5239, 50, 513anbi123i 1142 . . . . . . 7
5349, 52bitri 241 . . . . . 6
5453biimpi 187 . . . . 5
55 nb3graprlem2 21461 . . . . 5 USGrph Neighbors Neighbors
5626, 48, 54, 55syl3an 1226 . . . 4 USGrph Neighbors Neighbors
5736, 45, 563anbi123d 1254 . . 3 USGrph Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
5815, 35, 573bitr2d 273 . 2 USGrph Neighbors Neighbors Neighbors
59 oveq2 6089 . . . . . 6 Neighbors Neighbors
6059eqeq1d 2444 . . . . 5 Neighbors Neighbors
61602rexbidv 2748 . . . 4 Neighbors Neighbors
62 oveq2 6089 . . . . . 6 Neighbors Neighbors
6362eqeq1d 2444 . . . . 5 Neighbors Neighbors
64632rexbidv 2748 . . . 4 Neighbors Neighbors
65 oveq2 6089 . . . . . 6 Neighbors Neighbors
6665eqeq1d 2444 . . . . 5 Neighbors Neighbors
67662rexbidv 2748 . . . 4 Neighbors Neighbors
6861, 64, 67raltpg 3859 . . 3 Neighbors Neighbors Neighbors Neighbors
69683ad2ant1 978 . 2 USGrph Neighbors Neighbors Neighbors Neighbors
70 raleq 2904 . . . . 5 Neighbors Neighbors
7170bicomd 193 . . . 4 Neighbors Neighbors
7271adantr 452 . . 3 USGrph Neighbors Neighbors
73723ad2ant2 979 . 2 USGrph Neighbors Neighbors
7458, 69, 733bitr2d 273 1 USGrph Neighbors
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2599  wral 2705  wrex 2706   cdif 3317  csn 3814  cpr 3815  ctp 3816  cop 3817   class class class wbr 4212   crn 4879  (class class class)co 6081   USGrph cusg 21365   Neighbors cnbgra 21430 This theorem is referenced by:  cusgra3vnbpr  21474 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-hash 11619  df-usgra 21367  df-nbgra 21433
 Copyright terms: Public domain W3C validator