Theorem otkelins2kg 4253

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

Assertion

Ref	Expression
otkelins2kg	|- ((A e. V /\ B e. W /\ C e. T) -> (<<{{A}}, <<B, C>>>> e. Ins2k D <-> <<A, C>> e. D))

Proof of Theorem otkelins2kg

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

Step	Hyp	Ref	Expression
1		snex 4111	. . . 4 |- {{A}} e. _V
2		opkex 4113	. . . 4 |- <<B, C>> e. _V
3		opkelins2kg 4251	. . . 4 |- (({{A}} e. _V /\ <<B, C>> e. _V) -> (<<{{A}}, <<B, C>>>> e. Ins2k D <-> E.xE.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, z>> e. D)))
4	1, 2, 3	mp2an 653	. . 3 |- (<<{{A}}, <<B, C>>>> e. Ins2k D <-> E.xE.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, z>> e. D))
5		3anass 938	. . . . . . 7 |- (({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, z>> e. D) <-> ({{A}} = {{x}} /\ (<<B, C>> = <<y, z>> /\ <<x, z>> e. D)))
6		eqcom 2355	. . . . . . . . 9 |- ({{A}} = {{x}} <-> {{x}} = {{A}})
7		snex 4111	. . . . . . . . . . 11 |- {x} e. _V
8	7	sneqb 3876	. . . . . . . . . 10 |- ({{x}} = {{A}} <-> {x} = {A})
9		vex 2862	. . . . . . . . . . 11 |- x e. _V
10	9	sneqb 3876	. . . . . . . . . 10 |- ({x} = {A} <-> x = A)
11	8, 10	bitri 240	. . . . . . . . 9 |- ({{x}} = {{A}} <-> x = A)
12	6, 11	bitri 240	. . . . . . . 8 |- ({{A}} = {{x}} <-> x = A)
13	12	anbi1i 676	. . . . . . 7 |- (({{A}} = {{x}} /\ (<<B, C>> = <<y, z>> /\ <<x, z>> e. D)) <-> (x = A /\ (<<B, C>> = <<y, z>> /\ <<x, z>> e. D)))
14	5, 13	bitri 240	. . . . . 6 |- (({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, z>> e. D) <-> (x = A /\ (<<B, C>> = <<y, z>> /\ <<x, z>> e. D)))
15	14	2exbii 1583	. . . . 5 |- (E.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, z>> e. D) <-> E.yE.z(x = A /\ (<<B, C>> = <<y, z>> /\ <<x, z>> e. D)))
16		19.42vv 1907	. . . . 5 |- (E.yE.z(x = A /\ (<<B, C>> = <<y, z>> /\ <<x, z>> e. D)) <-> (x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, z>> e. D)))
17	15, 16	bitri 240	. . . 4 |- (E.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, z>> e. D) <-> (x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, z>> e. D)))
18	17	exbii 1582	. . 3 |- (E.xE.yE.z({{A}} = {{x}} /\ <<B, C>> = <<y, z>> /\ <<x, z>> e. D) <-> E.x(x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, z>> e. D)))
19	4, 18	bitri 240	. 2 |- (<<{{A}}, <<B, C>>>> e. Ins2k D <-> E.x(x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, z>> e. D)))
20		opkeq1 4059	. . . . . . . 8 |- (x = A -> <<x, z>> = <<A, z>>)
21	20	eleq1d 2419	. . . . . . 7 |- (x = A -> (<<x, z>> e. D <-> <<A, z>> e. D))
22	21	anbi2d 684	. . . . . 6 |- (x = A -> ((<<B, C>> = <<y, z>> /\ <<x, z>> e. D) <-> (<<B, C>> = <<y, z>> /\ <<A, z>> e. D)))
23	22	2exbidv 1628	. . . . 5 |- (x = A -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<x, z>> e. D) <-> E.yE.z(<<B, C>> = <<y, z>> /\ <<A, z>> e. D)))
24	23	ceqsexgv 2971	. . . 4 |- (A e. V -> (E.x(x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, z>> e. D)) <-> E.yE.z(<<B, C>> = <<y, z>> /\ <<A, z>> e. D)))
25	24	3ad2ant1 976	. . 3 |- ((A e. V /\ B e. W /\ C e. T) -> (E.x(x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, z>> e. D)) <-> E.yE.z(<<B, C>> = <<y, z>> /\ <<A, z>> e. D)))
26		eqcom 2355	. . . . . . . . . 10 |- (<<B, C>> = <<y, z>> <-> <<y, z>> = <<B, C>>)
27		vex 2862	. . . . . . . . . . 11 |- y e. _V
28		vex 2862	. . . . . . . . . . 11 |- z e. _V
29		opkthg 4131	. . . . . . . . . . 11 |- ((y e. _V /\ z e. _V /\ C e. T) -> (<<y, z>> = <<B, C>> <-> (y = B /\ z = C)))
30	27, 28, 29	mp3an12 1267	. . . . . . . . . 10 |- (C e. T -> (<<y, z>> = <<B, C>> <-> (y = B /\ z = C)))
31	26, 30	syl5bb 248	. . . . . . . . 9 |- (C e. T -> (<<B, C>> = <<y, z>> <-> (y = B /\ z = C)))
32	31	anbi1d 685	. . . . . . . 8 |- (C e. T -> ((<<B, C>> = <<y, z>> /\ <<A, z>> e. D) <-> ((y = B /\ z = C) /\ <<A, z>> e. D)))
33	32	2exbidv 1628	. . . . . . 7 |- (C e. T -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, z>> e. D) <-> E.yE.z((y = B /\ z = C) /\ <<A, z>> e. D)))
34		anass 630	. . . . . . . . . 10 |- (((y = B /\ z = C) /\ <<A, z>> e. D) <-> (y = B /\ (z = C /\ <<A, z>> e. D)))
35	34	exbii 1582	. . . . . . . . 9 |- (E.z((y = B /\ z = C) /\ <<A, z>> e. D) <-> E.z(y = B /\ (z = C /\ <<A, z>> e. D)))
36		19.42v 1905	. . . . . . . . 9 |- (E.z(y = B /\ (z = C /\ <<A, z>> e. D)) <-> (y = B /\ E.z(z = C /\ <<A, z>> e. D)))
37	35, 36	bitri 240	. . . . . . . 8 |- (E.z((y = B /\ z = C) /\ <<A, z>> e. D) <-> (y = B /\ E.z(z = C /\ <<A, z>> e. D)))
38	37	exbii 1582	. . . . . . 7 |- (E.yE.z((y = B /\ z = C) /\ <<A, z>> e. D) <-> E.y(y = B /\ E.z(z = C /\ <<A, z>> e. D)))
39	33, 38	syl6bb 252	. . . . . 6 |- (C e. T -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, z>> e. D) <-> E.y(y = B /\ E.z(z = C /\ <<A, z>> e. D))))
40	39	adantl 452	. . . . 5 |- ((B e. W /\ C e. T) -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, z>> e. D) <-> E.y(y = B /\ E.z(z = C /\ <<A, z>> e. D))))
41		biidd 228	. . . . . . 7 |- (y = B -> (E.z(z = C /\ <<A, z>> e. D) <-> E.z(z = C /\ <<A, z>> e. D)))
42	41	ceqsexgv 2971	. . . . . 6 |- (B e. W -> (E.y(y = B /\ E.z(z = C /\ <<A, z>> e. D)) <-> E.z(z = C /\ <<A, z>> e. D)))
43		opkeq2 4060	. . . . . . . 8 |- (z = C -> <<A, z>> = <<A, C>>)
44	43	eleq1d 2419	. . . . . . 7 |- (z = C -> (<<A, z>> e. D <-> <<A, C>> e. D))
45	44	ceqsexgv 2971	. . . . . 6 |- (C e. T -> (E.z(z = C /\ <<A, z>> e. D) <-> <<A, C>> e. D))
46	42, 45	sylan9bb 680	. . . . 5 |- ((B e. W /\ C e. T) -> (E.y(y = B /\ E.z(z = C /\ <<A, z>> e. D)) <-> <<A, C>> e. D))
47	40, 46	bitrd 244	. . . 4 |- ((B e. W /\ C e. T) -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, z>> e. D) <-> <<A, C>> e. D))
48	47	3adant1 973	. . 3 |- ((A e. V /\ B e. W /\ C e. T) -> (E.yE.z(<<B, C>> = <<y, z>> /\ <<A, z>> e. D) <-> <<A, C>> e. D))
49	25, 48	bitrd 244	. 2 |- ((A e. V /\ B e. W /\ C e. T) -> (E.x(x = A /\ E.yE.z(<<B, C>> = <<y, z>> /\ <<x, z>> e. D)) <-> <<A, C>> e. D))
50	19, 49	syl5bb 248	1 |- ((A e. V /\ B e. W /\ C e. T) -> (<<{{A}}, <<B, C>>>> e. Ins2k D <-> <<A, C>> e. D))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859  {csn 3737  <<copk 4057   Ins2_k cins2k 4176

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 4078  ax-sn 4087

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-ins2k 4187

This theorem is referenced by:  otkelins2k  4255  opkelimagekg  4271