Theorem otkelins3kg 4255

Description: Kuratowski ordered triple membership in Kuratowski insertion operator. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
otkelins3kg	⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ T) → (⟪{{A}}, ⟪B, C⟫⟫ ∈ Ins3k D ↔ ⟪A, B⟫ ∈ D))

Proof of Theorem otkelins3kg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4112	. . 3 ⊢ {{A}} ∈ V
2		opkex 4114	. . 3 ⊢ ⟪B, C⟫ ∈ V
3		opkelins3kg 4253	. . 3 ⊢ (({{A}} ∈ V ∧ ⟪B, C⟫ ∈ V) → (⟪{{A}}, ⟪B, C⟫⟫ ∈ Ins3k D ↔ ∃x∃y∃z({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)))
4	1, 2, 3	mp2an 653	. 2 ⊢ (⟪{{A}}, ⟪B, C⟫⟫ ∈ Ins3k D ↔ ∃x∃y∃z({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D))
5		3anass 938	. . . . . . . . 9 ⊢ (({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ ({{A}} = {{x}} ∧ (⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)))
6		eqcom 2355	. . . . . . . . . . 11 ⊢ ({{A}} = {{x}} ↔ {{x}} = {{A}})
7		snex 4112	. . . . . . . . . . . . 13 ⊢ {x} ∈ V
8	7	sneqb 3877	. . . . . . . . . . . 12 ⊢ ({{x}} = {{A}} ↔ {x} = {A})
9		vex 2863	. . . . . . . . . . . . 13 ⊢ x ∈ V
10	9	sneqb 3877	. . . . . . . . . . . 12 ⊢ ({x} = {A} ↔ x = A)
11	8, 10	bitri 240	. . . . . . . . . . 11 ⊢ ({{x}} = {{A}} ↔ x = A)
12	6, 11	bitri 240	. . . . . . . . . 10 ⊢ ({{A}} = {{x}} ↔ x = A)
13	12	anbi1i 676	. . . . . . . . 9 ⊢ (({{A}} = {{x}} ∧ (⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)) ↔ (x = A ∧ (⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)))
14	5, 13	bitri 240	. . . . . . . 8 ⊢ (({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ (x = A ∧ (⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)))
15	14	2exbii 1583	. . . . . . 7 ⊢ (∃y∃z({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ ∃y∃z(x = A ∧ (⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)))
16		19.42vv 1907	. . . . . . 7 ⊢ (∃y∃z(x = A ∧ (⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)) ↔ (x = A ∧ ∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)))
17	15, 16	bitri 240	. . . . . 6 ⊢ (∃y∃z({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ (x = A ∧ ∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)))
18	17	exbii 1582	. . . . 5 ⊢ (∃x∃y∃z({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ ∃x(x = A ∧ ∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)))
19		opkeq1 4060	. . . . . . . . 9 ⊢ (x = A → ⟪x, y⟫ = ⟪A, y⟫)
20	19	eleq1d 2419	. . . . . . . 8 ⊢ (x = A → (⟪x, y⟫ ∈ D ↔ ⟪A, y⟫ ∈ D))
21	20	anbi2d 684	. . . . . . 7 ⊢ (x = A → ((⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ (⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D)))
22	21	2exbidv 1628	. . . . . 6 ⊢ (x = A → (∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ ∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D)))
23	22	ceqsexgv 2972	. . . . 5 ⊢ (A ∈ V → (∃x(x = A ∧ ∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D)) ↔ ∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D)))
24	18, 23	syl5bb 248	. . . 4 ⊢ (A ∈ V → (∃x∃y∃z({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ ∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D)))
25	24	3ad2ant1 976	. . 3 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ T) → (∃x∃y∃z({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ ∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D)))
26		eqcom 2355	. . . . . . . . . . 11 ⊢ (⟪B, C⟫ = ⟪y, z⟫ ↔ ⟪y, z⟫ = ⟪B, C⟫)
27		vex 2863	. . . . . . . . . . . 12 ⊢ y ∈ V
28		vex 2863	. . . . . . . . . . . 12 ⊢ z ∈ V
29		opkthg 4132	. . . . . . . . . . . 12 ⊢ ((y ∈ V ∧ z ∈ V ∧ C ∈ T) → (⟪y, z⟫ = ⟪B, C⟫ ↔ (y = B ∧ z = C)))
30	27, 28, 29	mp3an12 1267	. . . . . . . . . . 11 ⊢ (C ∈ T → (⟪y, z⟫ = ⟪B, C⟫ ↔ (y = B ∧ z = C)))
31	26, 30	syl5bb 248	. . . . . . . . . 10 ⊢ (C ∈ T → (⟪B, C⟫ = ⟪y, z⟫ ↔ (y = B ∧ z = C)))
32	31	anbi1d 685	. . . . . . . . 9 ⊢ (C ∈ T → ((⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D) ↔ ((y = B ∧ z = C) ∧ ⟪A, y⟫ ∈ D)))
33		anass 630	. . . . . . . . 9 ⊢ (((y = B ∧ z = C) ∧ ⟪A, y⟫ ∈ D) ↔ (y = B ∧ (z = C ∧ ⟪A, y⟫ ∈ D)))
34	32, 33	syl6bb 252	. . . . . . . 8 ⊢ (C ∈ T → ((⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D) ↔ (y = B ∧ (z = C ∧ ⟪A, y⟫ ∈ D))))
35	34	2exbidv 1628	. . . . . . 7 ⊢ (C ∈ T → (∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D) ↔ ∃y∃z(y = B ∧ (z = C ∧ ⟪A, y⟫ ∈ D))))
36		exdistr 1906	. . . . . . 7 ⊢ (∃y∃z(y = B ∧ (z = C ∧ ⟪A, y⟫ ∈ D)) ↔ ∃y(y = B ∧ ∃z(z = C ∧ ⟪A, y⟫ ∈ D)))
37	35, 36	syl6bb 252	. . . . . 6 ⊢ (C ∈ T → (∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D) ↔ ∃y(y = B ∧ ∃z(z = C ∧ ⟪A, y⟫ ∈ D))))
38	37	adantl 452	. . . . 5 ⊢ ((B ∈ W ∧ C ∈ T) → (∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D) ↔ ∃y(y = B ∧ ∃z(z = C ∧ ⟪A, y⟫ ∈ D))))
39		opkeq2 4061	. . . . . . . . . 10 ⊢ (y = B → ⟪A, y⟫ = ⟪A, B⟫)
40	39	eleq1d 2419	. . . . . . . . 9 ⊢ (y = B → (⟪A, y⟫ ∈ D ↔ ⟪A, B⟫ ∈ D))
41	40	anbi2d 684	. . . . . . . 8 ⊢ (y = B → ((z = C ∧ ⟪A, y⟫ ∈ D) ↔ (z = C ∧ ⟪A, B⟫ ∈ D)))
42	41	exbidv 1626	. . . . . . 7 ⊢ (y = B → (∃z(z = C ∧ ⟪A, y⟫ ∈ D) ↔ ∃z(z = C ∧ ⟪A, B⟫ ∈ D)))
43	42	ceqsexgv 2972	. . . . . 6 ⊢ (B ∈ W → (∃y(y = B ∧ ∃z(z = C ∧ ⟪A, y⟫ ∈ D)) ↔ ∃z(z = C ∧ ⟪A, B⟫ ∈ D)))
44		biidd 228	. . . . . . 7 ⊢ (z = C → (⟪A, B⟫ ∈ D ↔ ⟪A, B⟫ ∈ D))
45	44	ceqsexgv 2972	. . . . . 6 ⊢ (C ∈ T → (∃z(z = C ∧ ⟪A, B⟫ ∈ D) ↔ ⟪A, B⟫ ∈ D))
46	43, 45	sylan9bb 680	. . . . 5 ⊢ ((B ∈ W ∧ C ∈ T) → (∃y(y = B ∧ ∃z(z = C ∧ ⟪A, y⟫ ∈ D)) ↔ ⟪A, B⟫ ∈ D))
47	38, 46	bitrd 244	. . . 4 ⊢ ((B ∈ W ∧ C ∈ T) → (∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D) ↔ ⟪A, B⟫ ∈ D))
48	47	3adant1 973	. . 3 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ T) → (∃y∃z(⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪A, y⟫ ∈ D) ↔ ⟪A, B⟫ ∈ D))
49	25, 48	bitrd 244	. 2 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ T) → (∃x∃y∃z({{A}} = {{x}} ∧ ⟪B, C⟫ = ⟪y, z⟫ ∧ ⟪x, y⟫ ∈ D) ↔ ⟪A, B⟫ ∈ D))
50	4, 49	syl5bb 248	1 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ T) → (⟪{{A}}, ⟪B, C⟫⟫ ∈ Ins3k D ↔ ⟪A, B⟫ ∈ D))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {csn 3738  ⟪copk 4058   Ins3_k cins3k 4178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-ins3k 4189

This theorem is referenced by:  otkelins3k  4257  opkelcokg  4262  opkelimagekg  4272