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Theorem 2pwuninel 7291
 Description: The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by NM, 27-Jun-2008.)
Assertion
Ref Expression
2pwuninel

Proof of Theorem 2pwuninel
StepHypRef Expression
1 sdomirr 7273 . . 3
2 elssuni 4067 . . . 4
3 ssdomg 7182 . . . . 5
4 canth2g 7290 . . . . . . 7
5 pwexb 4782 . . . . . . . 8
6 canth2g 7290 . . . . . . . 8
75, 6sylbi 189 . . . . . . 7
8 sdomtr 7274 . . . . . . 7
94, 7, 8syl2anc 644 . . . . . 6
10 domsdomtr 7271 . . . . . . 7
1110ex 425 . . . . . 6
129, 11syl5com 29 . . . . 5
133, 12syld 43 . . . 4
142, 13syl5 31 . . 3
151, 14mtoi 172 . 2
16 elex 2970 . . . 4
17 pwexb 4782 . . . . 5
185, 17bitri 242 . . . 4
1916, 18sylibr 205 . . 3
2019con3i 130 . 2
2115, 20pm2.61i 159 1
 Colors of variables: wff set class Syntax hints:   wn 3   wcel 1727  cvv 2962   wss 3306  cpw 3823  cuni 4039   class class class wbr 4237   cdom 7136   csdm 7137 This theorem is referenced by:  mnfnre  9159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141
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