Theorem ins2kss 4280

Description: Subset law for Ins2_k A. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
ins2kss	|- Ins2k A C_ (~P1 1c X.k (_V X.k _V))

Proof of Theorem ins2kss

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

Step	Hyp	Ref	Expression
1		vex 2863	. . . . 5 |- y e. _V
2		vex 2863	. . . . 5 |- z e. _V
3		opkelins2kg 4252	. . . . 5 |- ((y e. _V /\ z e. _V) -> (<<y, z>> e. Ins2k A <-> E.wE.tE.u(y = {{w}} /\ z = <<t, u>> /\ <<w, u>> e. A)))
4	1, 2, 3	mp2an 653	. . . 4 |- (<<y, z>> e. Ins2k A <-> E.wE.tE.u(y = {{w}} /\ z = <<t, u>> /\ <<w, u>> e. A))
5		opkeq12 4062	. . . . . . . 8 |- ((y = {{w}} /\ z = <<t, u>>) -> <<y, z>> = <<{{w}}, <<t, u>>>>)
6		vex 2863	. . . . . . . . . . 11 |- w e. _V
7	6	snel1c 4141	. . . . . . . . . 10 |- {w} e. 1c
8		snelpw1 4147	. . . . . . . . . 10 |- ({{w}} e. ~P1 1c <-> {w} e. 1c)
9	7, 8	mpbir 200	. . . . . . . . 9 |- {{w}} e. ~P1 1c
10		vex 2863	. . . . . . . . . 10 |- t e. _V
11		vex 2863	. . . . . . . . . 10 |- u e. _V
12	10, 11	opkelxpk 4249	. . . . . . . . . 10 |- (<<t, u>> e. (_V X.k _V) <-> (t e. _V /\ u e. _V))
13	10, 11, 12	mpbir2an 886	. . . . . . . . 9 |- <<t, u>> e. (_V X.k _V)
14		snex 4112	. . . . . . . . . 10 |- {{w}} e. _V
15		opkex 4114	. . . . . . . . . 10 |- <<t, u>> e. _V
16	14, 15	opkelxpk 4249	. . . . . . . . 9 |- (<<{{w}}, <<t, u>>>> e. (~P1 1c X.k (_V X.k _V)) <-> ({{w}} e. ~P1 1c /\ <<t, u>> e. (_V X.k _V)))
17	9, 13, 16	mpbir2an 886	. . . . . . . 8 |- <<{{w}}, <<t, u>>>> e. (~P1 1c X.k (_V X.k _V))
18	5, 17	syl6eqel 2441	. . . . . . 7 |- ((y = {{w}} /\ z = <<t, u>>) -> <<y, z>> e. (~P1 1c X.k (_V X.k _V)))
19	18	3adant3 975	. . . . . 6 |- ((y = {{w}} /\ z = <<t, u>> /\ <<w, u>> e. A) -> <<y, z>> e. (~P1 1c X.k (_V X.k _V)))
20	19	exlimiv 1634	. . . . 5 |- (E.u(y = {{w}} /\ z = <<t, u>> /\ <<w, u>> e. A) -> <<y, z>> e. (~P1 1c X.k (_V X.k _V)))
21	20	exlimivv 1635	. . . 4 |- (E.wE.tE.u(y = {{w}} /\ z = <<t, u>> /\ <<w, u>> e. A) -> <<y, z>> e. (~P1 1c X.k (_V X.k _V)))
22	4, 21	sylbi 187	. . 3 |- (<<y, z>> e. Ins2k A -> <<y, z>> e. (~P1 1c X.k (_V X.k _V)))
23	22	gen2 1547	. 2 |- A.yA.z(<<y, z>> e. Ins2k A -> <<y, z>> e. (~P1 1c X.k (_V X.k _V)))
24		df-ins2k 4188	. . . 4 |- Ins2k A = {x | E.yE.z(x = <<y, z>> /\ E.tE.uE.w(y = {{t}} /\ z = <<u, w>> /\ <<t, w>> e. A))}
25	24	opkabssvvki 4210	. . 3 |- Ins2k A C_ (_V X.k _V)
26		ssrelk 4212	. . 3 |- ( Ins2k A C_ (_V X.k _V) -> ( Ins2k A C_ (~P1 1c X.k (_V X.k _V)) <-> A.yA.z(<<y, z>> e. Ins2k A -> <<y, z>> e. (~P1 1c X.k (_V X.k _V)))))
27	25, 26	ax-mp 5	. 2 |- ( Ins2k A C_ (~P1 1c X.k (_V X.k _V)) <-> A.yA.z(<<y, z>> e. Ins2k A -> <<y, z>> e. (~P1 1c X.k (_V X.k _V))))
28	23, 27	mpbir 200	1 |- Ins2k A C_ (~P1 1c X.k (_V X.k _V))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860   C_ wss 3258  {csn 3738  <<copk 4058  1_cc1c 4135  ~P₁ cpw1 4136   X._k cxpk 4175   Ins2_k cins2k 4177

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-ral 2620  df-rex 2621  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-pw 3725  df-sn 3742  df-pr 3743  df-opk 4059  df-1c 4137  df-pw1 4138  df-xpk 4186  df-ins2k 4188

This theorem is referenced by:  ins2kexg  4306