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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 Sk 1 k k1c

Proof of Theorem dfpw2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . . 7
21elimak 4259 . . . . . 6 Sk 1 k k1c 1c Sk 1 k
3 el1c 4139 . . . . . . . . . 10 1c
43anbi1i 676 . . . . . . . . 9 1c Sk 1 k Sk 1 k
5 19.41v 1901 . . . . . . . . 9 Sk 1 k Sk 1 k
64, 5bitr4i 243 . . . . . . . 8 1c Sk 1 k Sk 1 k
76exbii 1582 . . . . . . 7 1c Sk 1 k Sk 1 k
8 df-rex 2620 . . . . . . 7 1c Sk 1 k 1c Sk 1 k
9 excom 1741 . . . . . . 7 Sk 1 k Sk 1 k
107, 8, 93bitr4i 268 . . . . . 6 1c Sk 1 k Sk 1 k
112, 10bitri 240 . . . . 5 Sk 1 k k1c Sk 1 k
12 snex 4111 . . . . . . . 8
13 opkeq1 4059 . . . . . . . . 9
1413eleq1d 2419 . . . . . . . 8 Sk 1 k Sk 1 k
1512, 14ceqsexv 2894 . . . . . . 7 Sk 1 k Sk 1 k
16 eldif 3221 . . . . . . . 8 Sk 1 k Sk 1 k
17 vex 2862 . . . . . . . . . 10
1817, 1elssetk 4270 . . . . . . . . 9 Sk
1912, 1opkelxpk 4248 . . . . . . . . . . . 12 1 k 1
201, 19mpbiran2 885 . . . . . . . . . . 11 1 k 1
21 snelpw1 4146 . . . . . . . . . . 11 1
2220, 21bitri 240 . . . . . . . . . 10 1 k
2322notbii 287 . . . . . . . . 9 1 k
2418, 23anbi12i 678 . . . . . . . 8 Sk 1 k
25 annim 414 . . . . . . . 8
2616, 24, 253bitri 262 . . . . . . 7 Sk 1 k
2715, 26bitri 240 . . . . . 6 Sk 1 k
2827exbii 1582 . . . . 5 Sk 1 k
29 exnal 1574 . . . . 5
3011, 28, 293bitri 262 . . . 4 Sk 1 k k1c
3130con2bii 322 . . 3 Sk 1 k k1c
321elpw 3728 . . . 4
33 dfss2 3262 . . . 4
3432, 33bitri 240 . . 3
351elcompl 3225 . . 3 Sk 1 k k1c Sk 1 k k1c
3631, 34, 353bitr4i 268 . 2 Sk 1 k k1c
3736eqriv 2350 1 Sk 1 k k1c
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 358  wal 1540  wex 1541   wceq 1642   wcel 1710  wrex 2615  cvv 2859   ∼ ccompl 3205   cdif 3206   wss 3257  cpw 3722  csn 3737  copk 4057  1cc1c 4134  1 cpw1 4135   k cxpk 4174  kcimak 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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