Theorem eqpwrelk 4479

Description: Represent equality to power class via a Kuratowski relationship. (Contributed by SF, 26-Jan-2015.)

Hypotheses

Ref	Expression
eqpwrelk.1	|- A e. _V
eqpwrelk.2	|- B e. _V

Assertion

Ref	Expression
eqpwrelk	|- (<<{A}, B>> e. ∼ (( Ins2k Sk (+) Ins3k SIk Sk )"k~P1 ~P1 1c) <-> B = ~PA)

Proof of Theorem eqpwrelk

Dummy variables x t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opkex 4114	. . . . 5 |- <<{A}, B>> e. _V
2	1	elimak 4260	. . . 4 |- (<<{A}, B>> e. (( Ins2k Sk (+) Ins3k SIk Sk )"k~P1 ~P1 1c) <-> E.t e. ~P1 ~P1 1c<<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk ))
3		elpw121c 4149	. . . . . . . 8 |- (t e. ~P1 ~P1 1c <-> E.x t = {{{x}}})
4	3	anbi1i 676	. . . . . . 7 |- ((t e. ~P1 ~P1 1c /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )) <-> (E.x t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )))
5		19.41v 1901	. . . . . . 7 |- (E.x(t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )) <-> (E.x t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )))
6	4, 5	bitr4i 243	. . . . . 6 |- ((t e. ~P1 ~P1 1c /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )) <-> E.x(t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )))
7	6	exbii 1582	. . . . 5 |- (E.t(t e. ~P1 ~P1 1c /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )) <-> E.tE.x(t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )))
8		df-rex 2621	. . . . 5 |- (E.t e. ~P1 ~P1 1c<<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk ) <-> E.t(t e. ~P1 ~P1 1c /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )))
9		excom 1741	. . . . 5 |- (E.xE.t(t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )) <-> E.tE.x(t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )))
10	7, 8, 9	3bitr4i 268	. . . 4 |- (E.t e. ~P1 ~P1 1c<<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk ) <-> E.xE.t(t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )))
11		snex 4112	. . . . . . 7 |- {{{x}}} e. _V
12		opkeq1 4060	. . . . . . . 8 |- (t = {{{x}}} -> <<t, <<{A}, B>>>> = <<{{{x}}}, <<{A}, B>>>>)
13	12	eleq1d 2419	. . . . . . 7 |- (t = {{{x}}} -> (<<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk ) <-> <<{{{x}}}, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )))
14	11, 13	ceqsexv 2895	. . . . . 6 |- (E.t(t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )) <-> <<{{{x}}}, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk ))
15		elsymdif 3224	. . . . . 6 |- (<<{{{x}}}, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk ) <-> -. (<<{{{x}}}, <<{A}, B>>>> e. Ins2k Sk <-> <<{{{x}}}, <<{A}, B>>>> e. Ins3k SIk Sk ))
16		snex 4112	. . . . . . . . . 10 |- {x} e. _V
17		snex 4112	. . . . . . . . . 10 |- {A} e. _V
18		eqpwrelk.2	. . . . . . . . . 10 |- B e. _V
19	16, 17, 18	otkelins2k 4256	. . . . . . . . 9 |- (<<{{{x}}}, <<{A}, B>>>> e. Ins2k Sk <-> <<{x}, B>> e. Sk )
20		vex 2863	. . . . . . . . . 10 |- x e. _V
21	20, 18	elssetk 4271	. . . . . . . . 9 |- (<<{x}, B>> e. Sk <-> x e. B)
22	19, 21	bitri 240	. . . . . . . 8 |- (<<{{{x}}}, <<{A}, B>>>> e. Ins2k Sk <-> x e. B)
23	16, 17, 18	otkelins3k 4257	. . . . . . . . 9 |- (<<{{{x}}}, <<{A}, B>>>> e. Ins3k SIk Sk <-> <<{x}, {A}>> e. SIk Sk )
24		eqpwrelk.1	. . . . . . . . . 10 |- A e. _V
25	20, 24	opksnelsik 4266	. . . . . . . . 9 |- (<<{x}, {A}>> e. SIk Sk <-> <<x, A>> e. Sk )
26		opkelssetkg 4269	. . . . . . . . . 10 |- ((x e. _V /\ A e. _V) -> (<<x, A>> e. Sk <-> x C_ A))
27	20, 24, 26	mp2an 653	. . . . . . . . 9 |- (<<x, A>> e. Sk <-> x C_ A)
28	23, 25, 27	3bitri 262	. . . . . . . 8 |- (<<{{{x}}}, <<{A}, B>>>> e. Ins3k SIk Sk <-> x C_ A)
29	22, 28	bibi12i 306	. . . . . . 7 |- ((<<{{{x}}}, <<{A}, B>>>> e. Ins2k Sk <-> <<{{{x}}}, <<{A}, B>>>> e. Ins3k SIk Sk ) <-> (x e. B <-> x C_ A))
30	29	notbii 287	. . . . . 6 |- (-. (<<{{{x}}}, <<{A}, B>>>> e. Ins2k Sk <-> <<{{{x}}}, <<{A}, B>>>> e. Ins3k SIk Sk ) <-> -. (x e. B <-> x C_ A))
31	14, 15, 30	3bitri 262	. . . . 5 |- (E.t(t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )) <-> -. (x e. B <-> x C_ A))
32	31	exbii 1582	. . . 4 |- (E.xE.t(t = {{{x}}} /\ <<t, <<{A}, B>>>> e. ( Ins2k Sk (+) Ins3k SIk Sk )) <-> E.x -. (x e. B <-> x C_ A))
33	2, 10, 32	3bitri 262	. . 3 |- (<<{A}, B>> e. (( Ins2k Sk (+) Ins3k SIk Sk )"k~P1 ~P1 1c) <-> E.x -. (x e. B <-> x C_ A))
34	33	notbii 287	. 2 |- (-. <<{A}, B>> e. (( Ins2k Sk (+) Ins3k SIk Sk )"k~P1 ~P1 1c) <-> -. E.x -. (x e. B <-> x C_ A))
35	1	elcompl 3226	. 2 |- (<<{A}, B>> e. ∼ (( Ins2k Sk (+) Ins3k SIk Sk )"k~P1 ~P1 1c) <-> -. <<{A}, B>> e. (( Ins2k Sk (+) Ins3k SIk Sk )"k~P1 ~P1 1c))
36		df-pw 3725	. . . 4 |- ~PA = {x | x C_ A}
37	36	eqeq2i 2363	. . 3 |- (B = ~PA <-> B = {x | x C_ A})
38		abeq2 2459	. . 3 |- (B = {x | x C_ A} <-> A.x(x e. B <-> x C_ A))
39		alex 1572	. . 3 |- (A.x(x e. B <-> x C_ A) <-> -. E.x -. (x e. B <-> x C_ A))
40	37, 38, 39	3bitri 262	. 2 |- (B = ~PA <-> -. E.x -. (x e. B <-> x C_ A))
41	34, 35, 40	3bitr4i 268	1 |- (<<{A}, B>> e. ∼ (( Ins2k Sk (+) Ins3k SIk Sk )"k~P1 ~P1 1c) <-> B = ~PA)

Syntax hints:  -. wn 3   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   (+) csymdif 3210   C_ wss 3258  ~Pcpw 3723  {csn 3738  <<copk 4058  1_cc1c 4135  ~P₁ cpw1 4136   Ins2_k cins2k 4177   Ins3_k cins3k 4178  "_kcimak 4180   SI_k csik 4182   S_k cssetk 4184

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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  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-ins2k 4188  df-ins3k 4189  df-imak 4190  df-sik 4193  df-ssetk 4194

This theorem is referenced by:  nnpweqlem1  4523