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Theorem nb3gr2nb 27094
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 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵})))

Proof of Theorem nb3gr2nb
StepHypRef Expression
1 prcom 4662 . . . . . . . . 9 {𝐴, 𝐶} = {𝐶, 𝐴}
21eleq1i 2903 . . . . . . . 8 ({𝐴, 𝐶} ∈ (Edg‘𝐺) ↔ {𝐶, 𝐴} ∈ (Edg‘𝐺))
32biimpi 217 . . . . . . 7 ({𝐴, 𝐶} ∈ (Edg‘𝐺) → {𝐶, 𝐴} ∈ (Edg‘𝐺))
43adantl 482 . . . . . 6 (({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) → {𝐶, 𝐴} ∈ (Edg‘𝐺))
5 prcom 4662 . . . . . . . . 9 {𝐵, 𝐶} = {𝐶, 𝐵}
65eleq1i 2903 . . . . . . . 8 ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ {𝐶, 𝐵} ∈ (Edg‘𝐺))
76biimpi 217 . . . . . . 7 ({𝐵, 𝐶} ∈ (Edg‘𝐺) → {𝐶, 𝐵} ∈ (Edg‘𝐺))
87adantl 482 . . . . . 6 (({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺)) → {𝐶, 𝐵} ∈ (Edg‘𝐺))
94, 8anim12i 612 . . . . 5 ((({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))) → ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺)))
109a1i 11 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))) → ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺))))
11 eqid 2821 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
12 eqid 2821 . . . . . 6 (Edg‘𝐺) = (Edg‘𝐺)
13 simprr 769 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
14 simprl 767 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶})
15 simpl 483 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (𝐴𝑋𝐵𝑌𝐶𝑍))
1611, 12, 13, 14, 15nb3grprlem1 27090 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺))))
17 3ancoma 1090 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) ↔ (𝐵𝑌𝐴𝑋𝐶𝑍))
1817biimpi 217 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐵𝑌𝐴𝑋𝐶𝑍))
19 tpcoma 4680 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
2019eqeq2i 2834 . . . . . . . 8 ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ↔ (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
2120biimpi 217 . . . . . . 7 ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} → (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
2221anim1i 614 . . . . . 6 (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph))
23 simprr 769 . . . . . . 7 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
24 simprl 767 . . . . . . 7 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
25 simpl 483 . . . . . . 7 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → (𝐵𝑌𝐴𝑋𝐶𝑍))
2611, 12, 23, 24, 25nb3grprlem1 27090 . . . . . 6 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ↔ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))))
2718, 22, 26syl2an 595 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ↔ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))))
2816, 27anbi12d 630 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ↔ (({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺)))))
29 3anrot 1092 . . . . . 6 ((𝐶𝑍𝐴𝑋𝐵𝑌) ↔ (𝐴𝑋𝐵𝑌𝐶𝑍))
3029biimpri 229 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐶𝑍𝐴𝑋𝐵𝑌))
31 tprot 4679 . . . . . . . . 9 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
3231eqcomi 2830 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
3332eqeq2i 2834 . . . . . . 7 ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ↔ (Vtx‘𝐺) = {𝐶, 𝐴, 𝐵})
3433anbi1i 623 . . . . . 6 (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ↔ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph))
3534biimpi 217 . . . . 5 (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph))
36 simprr 769 . . . . . 6 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
37 simprl 767 . . . . . 6 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐶, 𝐴, 𝐵})
38 simpl 483 . . . . . 6 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → (𝐶𝑍𝐴𝑋𝐵𝑌))
3911, 12, 36, 37, 38nb3grprlem1 27090 . . . . 5 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺))))
4030, 35, 39syl2an 595 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺))))
4110, 28, 403imtr4d 295 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) → (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}))
4241pm4.71d 562 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ↔ (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵})))
43 df-3an 1081 . 2 (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}) ↔ (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}))
4442, 43syl6bbr 290 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cpr 4561  {ctp 4563  cfv 6349  (class class class)co 7145  Vtxcvtx 26709  Edgcedg 26760  USGraphcusgr 26862   NeighbVtx cnbgr 27042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-fz 12883  df-hash 13681  df-edg 26761  df-upgr 26795  df-umgr 26796  df-usgr 26864  df-nbgr 27043
This theorem is referenced by: (None)
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