Theorem nb3gr2nb 27751

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	⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((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

Step	Hyp	Ref	Expression
1		prcom 4668	. . . . . . . . 9 ⊢ {𝐴, 𝐶} = {𝐶, 𝐴}
2	1	eleq1i 2829	. . . . . . . 8 ⊢ ({𝐴, 𝐶} ∈ (Edg‘𝐺) ↔ {𝐶, 𝐴} ∈ (Edg‘𝐺))
3	2	biimpi 215	. . . . . . 7 ⊢ ({𝐴, 𝐶} ∈ (Edg‘𝐺) → {𝐶, 𝐴} ∈ (Edg‘𝐺))
4	3	adantl 482	. . . . . 6 ⊢ (({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) → {𝐶, 𝐴} ∈ (Edg‘𝐺))
5		prcom 4668	. . . . . . . . 9 ⊢ {𝐵, 𝐶} = {𝐶, 𝐵}
6	5	eleq1i 2829	. . . . . . . 8 ⊢ ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ {𝐶, 𝐵} ∈ (Edg‘𝐺))
7	6	biimpi 215	. . . . . . 7 ⊢ ({𝐵, 𝐶} ∈ (Edg‘𝐺) → {𝐶, 𝐵} ∈ (Edg‘𝐺))
8	7	adantl 482	. . . . . 6 ⊢ (({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺)) → {𝐶, 𝐵} ∈ (Edg‘𝐺))
9	4, 8	anim12i 613	. . . . 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)))
10	9	a1i 11	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((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 2738	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		eqid 2738	. . . . . 6 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13		simprr 770	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
14		simprl 768	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶})
15		simpl 483	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
16	11, 12, 13, 14, 15	nb3grprlem1 27747	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺))))
17		3ancoma 1097	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ↔ (𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍))
18	17	biimpi 215	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍))
19		tpcoma 4686	. . . . . . . . 9 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
20	19	eqeq2i 2751	. . . . . . . 8 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ↔ (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
21	20	biimpi 215	. . . . . . 7 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} → (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
22	21	anim1i 615	. . . . . 6 ⊢ (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph))
23		simprr 770	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
24		simprl 768	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
25		simpl 483	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → (𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍))
26	11, 12, 23, 24, 25	nb3grprlem1 27747	. . . . . 6 ⊢ (((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ↔ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))))
27	18, 22, 26	syl2an 596	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ↔ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))))
28	16, 27	anbi12d 631	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((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 1099	. . . . . 6 ⊢ ((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍))
30	29	biimpri 227	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
31		tprot 4685	. . . . . . . . 9 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
32	31	eqcomi 2747	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
33	32	eqeq2i 2751	. . . . . . 7 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ↔ (Vtx‘𝐺) = {𝐶, 𝐴, 𝐵})
34	33	anbi1i 624	. . . . . 6 ⊢ (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ↔ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph))
35	34	biimpi 215	. . . . 5 ⊢ (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph))
36		simprr 770	. . . . . 6 ⊢ (((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
37		simprl 768	. . . . . 6 ⊢ (((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐶, 𝐴, 𝐵})
38		simpl 483	. . . . . 6 ⊢ (((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → (𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
39	11, 12, 36, 37, 38	nb3grprlem1 27747	. . . . 5 ⊢ (((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺))))
40	30, 35, 39	syl2an 596	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺))))
41	10, 28, 40	3imtr4d 294	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) → (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}))
42	41	pm4.71d 562	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((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 1088	. 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}))
44	42, 43	bitr4di 289	1 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((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})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cpr 4563  {ctp 4565  ‘cfv 6433  (class class class)co 7275  Vtxcvtx 27366  Edgcedg 27417  USGraphcusgr 27519   NeighbVtx cnbgr 27699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-edg 27418  df-upgr 27452  df-umgr 27453  df-usgr 27521  df-nbgr 27700

This theorem is referenced by: (None)