Theorem nb3gr2nb 29287

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 4692	. . . . . . . . 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 4692	. . . . . . . . 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 29283	. . . . 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 4710	. . . . . . . . 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 29283	. . . . . 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 4709	. . . . . . . . 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 29283	. . . . 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 4587  {ctp 4589  ‘cfv 6499  (class class class)co 7369  Vtxcvtx 28899  Edgcedg 28950  USGraphcusgr 29052   NeighbVtx cnbgr 29235

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9830  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-n0 12419  df-xnn0 12492  df-z 12506  df-uz 12770  df-fz 13445  df-hash 14272  df-edg 28951  df-upgr 28985  df-umgr 28986  df-usgr 29054  df-nbgr 29236

This theorem is referenced by: (None)