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Theorem pwexg 4328
 Description: The power class of a set is a set. (Contributed by SF, 21-Jan-2015.)
Assertion
Ref Expression
pwexg (A VA V)

Proof of Theorem pwexg
StepHypRef Expression
1 dfpw2 4327 . 2 A = ∼ (( Sk (1A ×k V)) “k 1c)
2 ssetkex 4294 . . . . 5 Sk V
3 pw1exg 4302 . . . . . 6 (A V1A V)
4 vvex 4109 . . . . . 6 V V
5 xpkexg 4288 . . . . . 6 ((1A V V V) → (1A ×k V) V)
63, 4, 5sylancl 643 . . . . 5 (A V → (1A ×k V) V)
7 difexg 4102 . . . . 5 (( Sk V (1A ×k V) V) → ( Sk (1A ×k V)) V)
82, 6, 7sylancr 644 . . . 4 (A V → ( Sk (1A ×k V)) V)
9 1cex 4142 . . . 4 1c V
10 imakexg 4299 . . . 4 ((( Sk (1A ×k V)) V 1c V) → (( Sk (1A ×k V)) “k 1c) V)
118, 9, 10sylancl 643 . . 3 (A V → (( Sk (1A ×k V)) “k 1c) V)
12 complexg 4099 . . 3 ((( Sk (1A ×k V)) “k 1c) V → ∼ (( Sk (1A ×k V)) “k 1c) V)
1311, 12syl 15 . 2 (A V → ∼ (( Sk (1A ×k V)) “k 1c) V)
141, 13syl5eqel 2437 1 (A VA V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206  ℘cpw 3722  1cc1c 4134  ℘1cpw1 4135   ×k cxpk 4174   “k cimak 4179   Sk cssetk 4183 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  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-ral 2619  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-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193 This theorem is referenced by:  pwex  4329  pmex  6005  ltcpw1pwg  6202
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