Theorem otkelins3kg 4255

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

Assertion

Ref	Expression
otkelins3kg	|- ((A e. V /\ B e. W /\ C e. T) -> (<<{{A}}, <<B, C>>>> e. Ins3k D <-> <<A, B>> e. 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}} e. _V
2		opkex 4114	. . 3 |- <<B, C>> e. _V
3		opkelins3kg 4253	. . 3 |- (({{A}} e. _V /\ <<B, C>> e. _V) -> (<<{{A}}, <<B, C>>>> e. Ins3k D <-> E.xE.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D)))
4	1, 2, 3	mp2an 653	. 2 |- (<<{{A}}, <<B, C>>>> e. Ins3k D <-> E.xE.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D))
5		3anass 938	. . . . . . . . 9 |- (({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> ({{A}} = {{x}} /\ (<<B, C>> = <<y, z>> /\ <<x, y>> e. D)))
6		eqcom 2355	. . . . . . . . . . 11 |- ({{A}} = {{x}} <-> {{x}} = {{A}})
7		snex 4112	. . . . . . . . . . . . 13 |- {x} e. _V
8	7	sneqb 3877	. . . . . . . . . . . 12 |- ({{x}} = {{A}} <-> {x} = {A})
9		vex 2863	. . . . . . . . . . . . 13 |- x e. _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>> e. D)) <-> (x = A /\ (<<B, C>> = <<y, z>> /\ <<x, y>> e. D)))
14	5, 13	bitri 240	. . . . . . . 8 |- (({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> (x = A /\ (<<B, C>> = <<y, z>> /\ <<x, y>> e. D)))
15	14	2exbii 1583	. . . . . . 7 |- (E.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> E.yE.z(x = A /\ (<<B, C>> = <<y, z>> /\ <<x, y>> e. D)))
16		19.42vv 1907	. . . . . . 7 |- (E.yE.z(x = A /\ (<<B, C>> = <<y, z>> /\ <<x, y>> e. D)) <-> (x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, y>> e. D)))
17	15, 16	bitri 240	. . . . . 6 |- (E.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> (x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, y>> e. D)))
18	17	exbii 1582	. . . . 5 |- (E.xE.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> E.x(x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, y>> e. D)))
19		opkeq1 4060	. . . . . . . . 9 |- (x = A -> <<x, y>> = <<A, y>>)
20	19	eleq1d 2419	. . . . . . . 8 |- (x = A -> (<<x, y>> e. D <-> <<A, y>> e. D))
21	20	anbi2d 684	. . . . . . 7 |- (x = A -> ((<<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> (<<B, C>> = <<y, z>> /\ <<A, y>> e. D)))
22	21	2exbidv 1628	. . . . . 6 |- (x = A -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> E.yE.z(<<B, C>> = <<y, z>> /\ <<A, y>> e. D)))
23	22	ceqsexgv 2972	. . . . 5 |- (A e. V -> (E.x(x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, y>> e. D)) <-> E.yE.z(<<B, C>> = <<y, z>> /\ <<A, y>> e. D)))
24	18, 23	syl5bb 248	. . . 4 |- (A e. V -> (E.xE.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> E.yE.z(<<B, C>> = <<y, z>> /\ <<A, y>> e. D)))
25	24	3ad2ant1 976	. . 3 |- ((A e. V /\ B e. W /\ C e. T) -> (E.xE.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> E.yE.z(<<B, C>> = <<y, z>> /\ <<A, y>> e. D)))
26		eqcom 2355	. . . . . . . . . . 11 |- (<<B, C>> = <<y, z>> <-> <<y, z>> = <<B, C>>)
27		vex 2863	. . . . . . . . . . . 12 |- y e. _V
28		vex 2863	. . . . . . . . . . . 12 |- z e. _V
29		opkthg 4132	. . . . . . . . . . . 12 |- ((y e. _V /\ z e. _V /\ C e. T) -> (<<y, z>> = <<B, C>> <-> (y = B /\ z = C)))
30	27, 28, 29	mp3an12 1267	. . . . . . . . . . 11 |- (C e. T -> (<<y, z>> = <<B, C>> <-> (y = B /\ z = C)))
31	26, 30	syl5bb 248	. . . . . . . . . 10 |- (C e. T -> (<<B, C>> = <<y, z>> <-> (y = B /\ z = C)))
32	31	anbi1d 685	. . . . . . . . 9 |- (C e. T -> ((<<B, C>> = <<y, z>> /\ <<A, y>> e. D) <-> ((y = B /\ z = C) /\ <<A, y>> e. D)))
33		anass 630	. . . . . . . . 9 |- (((y = B /\ z = C) /\ <<A, y>> e. D) <-> (y = B /\ (z = C /\ <<A, y>> e. D)))
34	32, 33	syl6bb 252	. . . . . . . 8 |- (C e. T -> ((<<B, C>> = <<y, z>> /\ <<A, y>> e. D) <-> (y = B /\ (z = C /\ <<A, y>> e. D))))
35	34	2exbidv 1628	. . . . . . 7 |- (C e. T -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, y>> e. D) <-> E.yE.z(y = B /\ (z = C /\ <<A, y>> e. D))))
36		exdistr 1906	. . . . . . 7 |- (E.yE.z(y = B /\ (z = C /\ <<A, y>> e. D)) <-> E.y(y = B /\ E.z(z = C /\ <<A, y>> e. D)))
37	35, 36	syl6bb 252	. . . . . 6 |- (C e. T -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, y>> e. D) <-> E.y(y = B /\ E.z(z = C /\ <<A, y>> e. D))))
38	37	adantl 452	. . . . 5 |- ((B e. W /\ C e. T) -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, y>> e. D) <-> E.y(y = B /\ E.z(z = C /\ <<A, y>> e. D))))
39		opkeq2 4061	. . . . . . . . . 10 |- (y = B -> <<A, y>> = <<A, B>>)
40	39	eleq1d 2419	. . . . . . . . 9 |- (y = B -> (<<A, y>> e. D <-> <<A, B>> e. D))
41	40	anbi2d 684	. . . . . . . 8 |- (y = B -> ((z = C /\ <<A, y>> e. D) <-> (z = C /\ <<A, B>> e. D)))
42	41	exbidv 1626	. . . . . . 7 |- (y = B -> (E.z(z = C /\ <<A, y>> e. D) <-> E.z(z = C /\ <<A, B>> e. D)))
43	42	ceqsexgv 2972	. . . . . 6 |- (B e. W -> (E.y(y = B /\ E.z(z = C /\ <<A, y>> e. D)) <-> E.z(z = C /\ <<A, B>> e. D)))
44		biidd 228	. . . . . . 7 |- (z = C -> (<<A, B>> e. D <-> <<A, B>> e. D))
45	44	ceqsexgv 2972	. . . . . 6 |- (C e. T -> (E.z(z = C /\ <<A, B>> e. D) <-> <<A, B>> e. D))
46	43, 45	sylan9bb 680	. . . . 5 |- ((B e. W /\ C e. T) -> (E.y(y = B /\ E.z(z = C /\ <<A, y>> e. D)) <-> <<A, B>> e. D))
47	38, 46	bitrd 244	. . . 4 |- ((B e. W /\ C e. T) -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, y>> e. D) <-> <<A, B>> e. D))
48	47	3adant1 973	. . 3 |- ((A e. V /\ B e. W /\ C e. T) -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, y>> e. D) <-> <<A, B>> e. D))
49	25, 48	bitrd 244	. 2 |- ((A e. V /\ B e. W /\ C e. T) -> (E.xE.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, y>> e. D) <-> <<A, B>> e. D))
50	4, 49	syl5bb 248	1 |- ((A e. V /\ B e. W /\ C e. T) -> (<<{{A}}, <<B, C>>>> e. Ins3k D <-> <<A, B>> e. D))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. 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