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Theorem dfpw2 4327
 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.
StepHypRef Expression
1 vex 2862 . . . . . . 7 x V
21elimak 4259 . . . . . 6 (x (( Sk (1A ×k V)) “k 1c) ↔ t 1ct, x ( Sk (1A ×k V)))
3 el1c 4139 . . . . . . . . . 10 (t 1cy t = {y})
43anbi1i 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))))
64, 5bitr4i 243 . . . . . . . 8 ((t 1c t, x ( Sk (1A ×k V))) ↔ y(t = {y} t, x ( Sk (1A ×k V))))
76exbii 1582 . . . . . . 7 (t(t 1c t, x ( Sk (1A ×k V))) ↔ ty(t = {y} t, x ( Sk (1A ×k V))))
8 df-rex 2620 . . . . . . 7 (t 1ct, x ( Sk (1A ×k V)) ↔ t(t 1c t, x ( Sk (1A ×k V))))
9 excom 1741 . . . . . . 7 (yt(t = {y} t, x ( Sk (1A ×k V))) ↔ ty(t = {y} t, x ( Sk (1A ×k V))))
107, 8, 93bitr4i 268 . . . . . 6 (t 1ct, x ( Sk (1A ×k V)) ↔ yt(t = {y} t, x ( Sk (1A ×k V))))
112, 10bitri 240 . . . . 5 (x (( Sk (1A ×k V)) “k 1c) ↔ yt(t = {y} t, x ( Sk (1A ×k V))))
12 snex 4111 . . . . . . . 8 {y} V
13 opkeq1 4059 . . . . . . . . 9 (t = {y} → ⟪t, x⟫ = ⟪{y}, x⟫)
1413eleq1d 2419 . . . . . . . 8 (t = {y} → (⟪t, x ( Sk (1A ×k V)) ↔ ⟪{y}, x ( Sk (1A ×k V))))
1512, 14ceqsexv 2894 . . . . . . 7 (t(t = {y} t, x ( Sk (1A ×k V))) ↔ ⟪{y}, x ( Sk (1A ×k V)))
16 eldif 3221 . . . . . . . 8 (⟪{y}, x ( Sk (1A ×k V)) ↔ (⟪{y}, x Sk ¬ ⟪{y}, x (1A ×k V)))
17 vex 2862 . . . . . . . . . 10 y V
1817, 1elssetk 4270 . . . . . . . . 9 (⟪{y}, x Sky x)
1912, 1opkelxpk 4248 . . . . . . . . . . . 12 (⟪{y}, x (1A ×k V) ↔ ({y} 1A x V))
201, 19mpbiran2 885 . . . . . . . . . . 11 (⟪{y}, x (1A ×k V) ↔ {y} 1A)
21 snelpw1 4146 . . . . . . . . . . 11 ({y} 1Ay A)
2220, 21bitri 240 . . . . . . . . . 10 (⟪{y}, x (1A ×k V) ↔ y A)
2322notbii 287 . . . . . . . . 9 (¬ ⟪{y}, x (1A ×k V) ↔ ¬ y A)
2418, 23anbi12i 678 . . . . . . . 8 ((⟪{y}, x Sk ¬ ⟪{y}, x (1A ×k V)) ↔ (y x ¬ y A))
25 annim 414 . . . . . . . 8 ((y x ¬ y A) ↔ ¬ (y xy A))
2616, 24, 253bitri 262 . . . . . . 7 (⟪{y}, x ( Sk (1A ×k V)) ↔ ¬ (y xy A))
2715, 26bitri 240 . . . . . 6 (t(t = {y} t, x ( Sk (1A ×k V))) ↔ ¬ (y xy A))
2827exbii 1582 . . . . 5 (yt(t = {y} t, x ( Sk (1A ×k V))) ↔ y ¬ (y xy A))
29 exnal 1574 . . . . 5 (y ¬ (y xy A) ↔ ¬ y(y xy A))
3011, 28, 293bitri 262 . . . 4 (x (( Sk (1A ×k V)) “k 1c) ↔ ¬ y(y xy A))
3130con2bii 322 . . 3 (y(y xy A) ↔ ¬ x (( Sk (1A ×k V)) “k 1c))
321elpw 3728 . . . 4 (x Ax A)
33 dfss2 3262 . . . 4 (x Ay(y xy A))
3432, 33bitri 240 . . 3 (x Ay(y xy A))
351elcompl 3225 . . 3 (x ∼ (( Sk (1A ×k V)) “k 1c) ↔ ¬ x (( Sk (1A ×k V)) “k 1c))
3631, 34, 353bitr4i 268 . 2 (x Ax ∼ (( Sk (1A ×k V)) “k 1c))
3736eqriv 2350 1 A = ∼ (( Sk (1A ×k V)) “k 1c)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206   ⊆ wss 3257  ℘cpw 3722  {csn 3737  ⟪copk 4057  1cc1c 4134  ℘1cpw1 4135   ×k cxpk 4174   “k cimak 4179   Sk cssetk 4183 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-rex 2620  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-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-imak 4189  df-ssetk 4193 This theorem is referenced by:  pwexg  4328
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