Theorem dfpw2 4328

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

Assertion

Ref	Expression
dfpw2	⊢ ℘A = ∼ (( Sk ∖ (℘1A ×k V)) “k 1c)

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 ∈ V
2	1	elimak 4260	. . . . . 6 ⊢ (x ∈ (( Sk ∖ (℘1A ×k V)) “k 1c) ↔ ∃t ∈ 1c ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V)))
3		el1c 4140	. . . . . . . . . 10 ⊢ (t ∈ 1c ↔ ∃y t = {y})
4	3	anbi1i 676	. . . . . . . . 9 ⊢ ((t ∈ 1c ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))) ↔ (∃y t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))))
5		19.41v 1901	. . . . . . . . 9 ⊢ (∃y(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))) ↔ (∃y t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))))
6	4, 5	bitr4i 243	. . . . . . . 8 ⊢ ((t ∈ 1c ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))) ↔ ∃y(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))))
7	6	exbii 1582	. . . . . . 7 ⊢ (∃t(t ∈ 1c ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))) ↔ ∃t∃y(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))))
8		df-rex 2621	. . . . . . 7 ⊢ (∃t ∈ 1c ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V)) ↔ ∃t(t ∈ 1c ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))))
9		excom 1741	. . . . . . 7 ⊢ (∃y∃t(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))) ↔ ∃t∃y(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))))
10	7, 8, 9	3bitr4i 268	. . . . . 6 ⊢ (∃t ∈ 1c ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V)) ↔ ∃y∃t(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))))
11	2, 10	bitri 240	. . . . 5 ⊢ (x ∈ (( Sk ∖ (℘1A ×k V)) “k 1c) ↔ ∃y∃t(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))))
12		snex 4112	. . . . . . . 8 ⊢ {y} ∈ V
13		opkeq1 4060	. . . . . . . . 9 ⊢ (t = {y} → ⟪t, x⟫ = ⟪{y}, x⟫)
14	13	eleq1d 2419	. . . . . . . 8 ⊢ (t = {y} → (⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V)) ↔ ⟪{y}, x⟫ ∈ ( Sk ∖ (℘1A ×k V))))
15	12, 14	ceqsexv 2895	. . . . . . 7 ⊢ (∃t(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))) ↔ ⟪{y}, x⟫ ∈ ( Sk ∖ (℘1A ×k V)))
16		eldif 3222	. . . . . . . 8 ⊢ (⟪{y}, x⟫ ∈ ( Sk ∖ (℘1A ×k V)) ↔ (⟪{y}, x⟫ ∈ Sk ∧ ¬ ⟪{y}, x⟫ ∈ (℘1A ×k V)))
17		vex 2863	. . . . . . . . . 10 ⊢ y ∈ V
18	17, 1	elssetk 4271	. . . . . . . . 9 ⊢ (⟪{y}, x⟫ ∈ Sk ↔ y ∈ x)
19	12, 1	opkelxpk 4249	. . . . . . . . . . . 12 ⊢ (⟪{y}, x⟫ ∈ (℘1A ×k V) ↔ ({y} ∈ ℘1A ∧ x ∈ V))
20	1, 19	mpbiran2 885	. . . . . . . . . . 11 ⊢ (⟪{y}, x⟫ ∈ (℘1A ×k V) ↔ {y} ∈ ℘1A)
21		snelpw1 4147	. . . . . . . . . . 11 ⊢ ({y} ∈ ℘1A ↔ y ∈ A)
22	20, 21	bitri 240	. . . . . . . . . 10 ⊢ (⟪{y}, x⟫ ∈ (℘1A ×k V) ↔ y ∈ A)
23	22	notbii 287	. . . . . . . . 9 ⊢ (¬ ⟪{y}, x⟫ ∈ (℘1A ×k V) ↔ ¬ y ∈ A)
24	18, 23	anbi12i 678	. . . . . . . 8 ⊢ ((⟪{y}, x⟫ ∈ Sk ∧ ¬ ⟪{y}, x⟫ ∈ (℘1A ×k V)) ↔ (y ∈ x ∧ ¬ y ∈ A))
25		annim 414	. . . . . . . 8 ⊢ ((y ∈ x ∧ ¬ y ∈ A) ↔ ¬ (y ∈ x → y ∈ A))
26	16, 24, 25	3bitri 262	. . . . . . 7 ⊢ (⟪{y}, x⟫ ∈ ( Sk ∖ (℘1A ×k V)) ↔ ¬ (y ∈ x → y ∈ A))
27	15, 26	bitri 240	. . . . . 6 ⊢ (∃t(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))) ↔ ¬ (y ∈ x → y ∈ A))
28	27	exbii 1582	. . . . 5 ⊢ (∃y∃t(t = {y} ∧ ⟪t, x⟫ ∈ ( Sk ∖ (℘1A ×k V))) ↔ ∃y ¬ (y ∈ x → y ∈ A))
29		exnal 1574	. . . . 5 ⊢ (∃y ¬ (y ∈ x → y ∈ A) ↔ ¬ ∀y(y ∈ x → y ∈ A))
30	11, 28, 29	3bitri 262	. . . 4 ⊢ (x ∈ (( Sk ∖ (℘1A ×k V)) “k 1c) ↔ ¬ ∀y(y ∈ x → y ∈ A))
31	30	con2bii 322	. . 3 ⊢ (∀y(y ∈ x → y ∈ A) ↔ ¬ x ∈ (( Sk ∖ (℘1A ×k V)) “k 1c))
32	1	elpw 3729	. . . 4 ⊢ (x ∈ ℘A ↔ x ⊆ A)
33		dfss2 3263	. . . 4 ⊢ (x ⊆ A ↔ ∀y(y ∈ x → y ∈ A))
34	32, 33	bitri 240	. . 3 ⊢ (x ∈ ℘A ↔ ∀y(y ∈ x → y ∈ A))
35	1	elcompl 3226	. . 3 ⊢ (x ∈ ∼ (( Sk ∖ (℘1A ×k V)) “k 1c) ↔ ¬ x ∈ (( Sk ∖ (℘1A ×k V)) “k 1c))
36	31, 34, 35	3bitr4i 268	. 2 ⊢ (x ∈ ℘A ↔ x ∈ ∼ (( Sk ∖ (℘1A ×k V)) “k 1c))
37	36	eqriv 2350	1 ⊢ ℘A = ∼ (( Sk ∖ (℘1A ×k V)) “k 1c)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ⊆ wss 3258  ℘cpw 3723  {csn 3738  ⟪copk 4058  1_cc1c 4135  ℘₁cpw1 4136   ×_k cxpk 4175   “_k cimak 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