Theorem nb3gr2nb 29311

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 4696	. . . . . . . . 9 ⊢ {𝐴, 𝐶} = {𝐶, 𝐴}
2	1	eleq1i 2819	. . . . . . . 8 ⊢ ({𝐴, 𝐶} ∈ (Edg‘𝐺) ↔ {𝐶, 𝐴} ∈ (Edg‘𝐺))
3	2	biimpi 216	. . . . . . 7 ⊢ ({𝐴, 𝐶} ∈ (Edg‘𝐺) → {𝐶, 𝐴} ∈ (Edg‘𝐺))
4	3	adantl 481	. . . . . 6 ⊢ (({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) → {𝐶, 𝐴} ∈ (Edg‘𝐺))
5		prcom 4696	. . . . . . . . 9 ⊢ {𝐵, 𝐶} = {𝐶, 𝐵}
6	5	eleq1i 2819	. . . . . . . 8 ⊢ ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ {𝐶, 𝐵} ∈ (Edg‘𝐺))
7	6	biimpi 216	. . . . . . 7 ⊢ ({𝐵, 𝐶} ∈ (Edg‘𝐺) → {𝐶, 𝐵} ∈ (Edg‘𝐺))
8	7	adantl 481	. . . . . 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 2729	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		eqid 2729	. . . . . 6 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13		simprr 772	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
14		simprl 770	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶})
15		simpl 482	. . . . . 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 29307	. . . . 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 216	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍))
19		tpcoma 4714	. . . . . . . . 9 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
20	19	eqeq2i 2742	. . . . . . . 8 ⊢ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ↔ (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
21	20	biimpi 216	. . . . . . 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 772	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
24		simprl 770	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
25		simpl 482	. . . . . . 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 29307	. . . . . 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 632	. . . 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 228	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌))
31		tprot 4713	. . . . . . . . 9 ⊢ {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
32	31	eqcomi 2738	. . . . . . . 8 ⊢ {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
33	32	eqeq2i 2742	. . . . . . 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 216	. . . . 5 ⊢ (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph))
36		simprr 772	. . . . . 6 ⊢ (((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
37		simprl 770	. . . . . 6 ⊢ (((𝐶 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐶, 𝐴, 𝐵})
38		simpl 482	. . . . . 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 29307	. . . . 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 561	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cpr 4591  {ctp 4593  ‘cfv 6511  (class class class)co 7387  Vtxcvtx 28923  Edgcedg 28974  USGraphcusgr 29076   NeighbVtx cnbgr 29259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-fz 13469  df-hash 14296  df-edg 28975  df-upgr 29009  df-umgr 29010  df-usgr 29078  df-nbgr 29260

This theorem is referenced by: (None)