Theorem dfpw2 4328

Description: Definition of power set for existence proof. (Contributed by SF, 21-Jan-2015.)

Assertion

Ref	Expression
dfpw2	|- ~PA = ∼ (( Sk \ (~P1 A X.k _V))"k1c)

Proof of Theorem dfpw2

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

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . 7 |- x e. _V
2	1	elimak 4260	. . . . . 6 |- (x e. (( Sk \ (~P1 A X.k _V))"k1c) <-> E.t e. 1c <<t, x>> e. ( Sk \ (~P1 A X.k _V)))
3		el1c 4140	. . . . . . . . . 10 |- (t e. 1c <-> E.y t = {y})
4	3	anbi1i 676	. . . . . . . . 9 |- ((t e. 1c /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))) <-> (E.y t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))))
5		19.41v 1901	. . . . . . . . 9 |- (E.y(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))) <-> (E.y t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))))
6	4, 5	bitr4i 243	. . . . . . . 8 |- ((t e. 1c /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))) <-> E.y(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))))
7	6	exbii 1582	. . . . . . 7 |- (E.t(t e. 1c /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))) <-> E.tE.y(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))))
8		df-rex 2621	. . . . . . 7 |- (E.t e. 1c <<t, x>> e. ( Sk \ (~P1 A X.k _V)) <-> E.t(t e. 1c /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))))
9		excom 1741	. . . . . . 7 |- (E.yE.t(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))) <-> E.tE.y(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))))
10	7, 8, 9	3bitr4i 268	. . . . . 6 |- (E.t e. 1c <<t, x>> e. ( Sk \ (~P1 A X.k _V)) <-> E.yE.t(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))))
11	2, 10	bitri 240	. . . . 5 |- (x e. (( Sk \ (~P1 A X.k _V))"k1c) <-> E.yE.t(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))))
12		snex 4112	. . . . . . . 8 |- {y} e. _V
13		opkeq1 4060	. . . . . . . . 9 |- (t = {y} -> <<t, x>> = <<{y}, x>>)
14	13	eleq1d 2419	. . . . . . . 8 |- (t = {y} -> (<<t, x>> e. ( Sk \ (~P1 A X.k _V)) <-> <<{y}, x>> e. ( Sk \ (~P1 A X.k _V))))
15	12, 14	ceqsexv 2895	. . . . . . 7 |- (E.t(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))) <-> <<{y}, x>> e. ( Sk \ (~P1 A X.k _V)))
16		eldif 3222	. . . . . . . 8 |- (<<{y}, x>> e. ( Sk \ (~P1 A X.k _V)) <-> (<<{y}, x>> e. Sk /\ -. <<{y}, x>> e. (~P1 A X.k _V)))
17		vex 2863	. . . . . . . . . 10 |- y e. _V
18	17, 1	elssetk 4271	. . . . . . . . 9 |- (<<{y}, x>> e. Sk <-> y e. x)
19	12, 1	opkelxpk 4249	. . . . . . . . . . . 12 |- (<<{y}, x>> e. (~P1 A X.k _V) <-> ({y} e. ~P1 A /\ x e. _V))
20	1, 19	mpbiran2 885	. . . . . . . . . . 11 |- (<<{y}, x>> e. (~P1 A X.k _V) <-> {y} e. ~P1 A)
21		snelpw1 4147	. . . . . . . . . . 11 |- ({y} e. ~P1 A <-> y e. A)
22	20, 21	bitri 240	. . . . . . . . . 10 |- (<<{y}, x>> e. (~P1 A X.k _V) <-> y e. A)
23	22	notbii 287	. . . . . . . . 9 |- (-. <<{y}, x>> e. (~P1 A X.k _V) <-> -. y e. A)
24	18, 23	anbi12i 678	. . . . . . . 8 |- ((<<{y}, x>> e. Sk /\ -. <<{y}, x>> e. (~P1 A X.k _V)) <-> (y e. x /\ -. y e. A))
25		annim 414	. . . . . . . 8 |- ((y e. x /\ -. y e. A) <-> -. (y e. x -> y e. A))
26	16, 24, 25	3bitri 262	. . . . . . 7 |- (<<{y}, x>> e. ( Sk \ (~P1 A X.k _V)) <-> -. (y e. x -> y e. A))
27	15, 26	bitri 240	. . . . . 6 |- (E.t(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))) <-> -. (y e. x -> y e. A))
28	27	exbii 1582	. . . . 5 |- (E.yE.t(t = {y} /\ <<t, x>> e. ( Sk \ (~P1 A X.k _V))) <-> E.y -. (y e. x -> y e. A))
29		exnal 1574	. . . . 5 |- (E.y -. (y e. x -> y e. A) <-> -. A.y(y e. x -> y e. A))
30	11, 28, 29	3bitri 262	. . . 4 |- (x e. (( Sk \ (~P1 A X.k _V))"k1c) <-> -. A.y(y e. x -> y e. A))
31	30	con2bii 322	. . 3 |- (A.y(y e. x -> y e. A) <-> -. x e. (( Sk \ (~P1 A X.k _V))"k1c))
32	1	elpw 3729	. . . 4 |- (x e. ~PA <-> x C_ A)
33		dfss2 3263	. . . 4 |- (x C_ A <-> A.y(y e. x -> y e. A))
34	32, 33	bitri 240	. . 3 |- (x e. ~PA <-> A.y(y e. x -> y e. A))
35	1	elcompl 3226	. . 3 |- (x e. ∼ (( Sk \ (~P1 A X.k _V))"k1c) <-> -. x e. (( Sk \ (~P1 A X.k _V))"k1c))
36	31, 34, 35	3bitr4i 268	. 2 |- (x e. ~PA <-> x e. ∼ (( Sk \ (~P1 A X.k _V))"k1c))
37	36	eqriv 2350	1 |- ~PA = ∼ (( Sk \ (~P1 A X.k _V))"k1c)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   \ cdif 3207   C_ wss 3258  ~Pcpw 3723  {csn 3738  <<copk 4058  1_cc1c 4135  ~P₁ cpw1 4136   X._k cxpk 4175  "_kcimak 4180   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-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-imak 4190  df-ssetk 4194

This theorem is referenced by:  pwexg  4329