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Theorem nb3gr2nb 39622
 Description: If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Assertion
Ref Expression
nb3gr2nb Vtx USGraph NeighbVtx NeighbVtx NeighbVtx NeighbVtx NeighbVtx

Proof of Theorem nb3gr2nb
StepHypRef Expression
1 prcom 4041 . . . . . . . . 9
21eleq1i 2540 . . . . . . . 8 Edg Edg
32biimpi 199 . . . . . . 7 Edg Edg
43adantl 473 . . . . . 6 Edg Edg Edg
5 prcom 4041 . . . . . . . . 9
65eleq1i 2540 . . . . . . . 8 Edg Edg
76biimpi 199 . . . . . . 7 Edg Edg
87adantl 473 . . . . . 6 Edg Edg Edg
94, 8anim12i 576 . . . . 5 Edg Edg Edg Edg Edg Edg
109a1i 11 . . . 4 Vtx USGraph Edg Edg Edg Edg Edg Edg
11 eqid 2471 . . . . . 6 Vtx Vtx
12 eqid 2471 . . . . . 6 Edg Edg
13 simprr 774 . . . . . 6 Vtx USGraph USGraph
14 simprl 772 . . . . . 6 Vtx USGraph Vtx
15 simpl 464 . . . . . 6 Vtx USGraph
1611, 12, 13, 14, 15nb3grprlem1 39618 . . . . 5 Vtx USGraph NeighbVtx Edg Edg
17 3ancoma 1014 . . . . . . 7
1817biimpi 199 . . . . . 6
19 tpcoma 4059 . . . . . . . . 9
2019eqeq2i 2483 . . . . . . . 8 Vtx Vtx
2120biimpi 199 . . . . . . 7 Vtx Vtx
2221anim1i 578 . . . . . 6 Vtx USGraph Vtx USGraph
23 simprr 774 . . . . . . 7 Vtx USGraph USGraph
24 simprl 772 . . . . . . 7 Vtx USGraph Vtx
25 simpl 464 . . . . . . 7 Vtx USGraph
2611, 12, 23, 24, 25nb3grprlem1 39618 . . . . . 6 Vtx USGraph NeighbVtx Edg Edg
2718, 22, 26syl2an 485 . . . . 5 Vtx USGraph NeighbVtx Edg Edg
2816, 27anbi12d 725 . . . 4 Vtx USGraph NeighbVtx NeighbVtx Edg Edg Edg Edg
29 3anrot 1012 . . . . . 6
3029biimpri 211 . . . . 5
31 tprot 4058 . . . . . . . . 9
3231eqcomi 2480 . . . . . . . 8
3332eqeq2i 2483 . . . . . . 7 Vtx Vtx
3433anbi1i 709 . . . . . 6 Vtx USGraph Vtx USGraph
3534biimpi 199 . . . . 5 Vtx USGraph Vtx USGraph
36 simprr 774 . . . . . 6 Vtx USGraph USGraph
37 simprl 772 . . . . . 6 Vtx USGraph Vtx
38 simpl 464 . . . . . 6 Vtx USGraph
3911, 12, 36, 37, 38nb3grprlem1 39618 . . . . 5 Vtx USGraph NeighbVtx Edg Edg
4030, 35, 39syl2an 485 . . . 4 Vtx USGraph NeighbVtx Edg Edg
4110, 28, 403imtr4d 276 . . 3 Vtx USGraph NeighbVtx NeighbVtx NeighbVtx
4241pm4.71d 646 . 2 Vtx USGraph NeighbVtx NeighbVtx NeighbVtx NeighbVtx NeighbVtx
43 df-3an 1009 . 2 NeighbVtx NeighbVtx NeighbVtx NeighbVtx NeighbVtx NeighbVtx
4442, 43syl6bbr 271 1 Vtx USGraph NeighbVtx NeighbVtx NeighbVtx NeighbVtx NeighbVtx
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   w3a 1007   wceq 1452   wcel 1904  cpr 3961  ctp 3963  cfv 5589  (class class class)co 6308  Vtxcvtx 39251  Edgcedga 39371   USGraph cusgr 39397   NeighbVtx cnbgr 39561 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-upgr 39328  df-umgr 39329  df-edga 39372  df-usgr 39399  df-nbgr 39565 This theorem is referenced by: (None)
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