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Theorem 2pwuninel 8156
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 ¬ 𝒫 𝒫 𝐴𝐴

Proof of Theorem 2pwuninel
StepHypRef Expression
1 sdomirr 8138 . . 3 ¬ 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴
2 elssuni 4499 . . . 4 (𝒫 𝒫 𝐴𝐴 → 𝒫 𝒫 𝐴 𝐴)
3 ssdomg 8043 . . . . 5 ( 𝐴 ∈ V → (𝒫 𝒫 𝐴 𝐴 → 𝒫 𝒫 𝐴 𝐴))
4 canth2g 8155 . . . . . 6 ( 𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
5 pwexb 7017 . . . . . . 7 ( 𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
6 canth2g 8155 . . . . . . 7 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴)
75, 6sylbi 207 . . . . . 6 ( 𝐴 ∈ V → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴)
8 sdomtr 8139 . . . . . 6 (( 𝐴 ≺ 𝒫 𝐴 ∧ 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) → 𝐴 ≺ 𝒫 𝒫 𝐴)
94, 7, 8syl2anc 694 . . . . 5 ( 𝐴 ∈ V → 𝐴 ≺ 𝒫 𝒫 𝐴)
10 domsdomtr 8136 . . . . . 6 ((𝒫 𝒫 𝐴 𝐴 𝐴 ≺ 𝒫 𝒫 𝐴) → 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴)
1110ex 449 . . . . 5 (𝒫 𝒫 𝐴 𝐴 → ( 𝐴 ≺ 𝒫 𝒫 𝐴 → 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴))
123, 9, 11syl6ci 71 . . . 4 ( 𝐴 ∈ V → (𝒫 𝒫 𝐴 𝐴 → 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴))
132, 12syl5 34 . . 3 ( 𝐴 ∈ V → (𝒫 𝒫 𝐴𝐴 → 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴))
141, 13mtoi 190 . 2 ( 𝐴 ∈ V → ¬ 𝒫 𝒫 𝐴𝐴)
15 elex 3243 . . . 4 (𝒫 𝒫 𝐴𝐴 → 𝒫 𝒫 𝐴 ∈ V)
16 pwexb 7017 . . . . 5 (𝒫 𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V)
175, 16bitri 264 . . . 4 ( 𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V)
1815, 17sylibr 224 . . 3 (𝒫 𝒫 𝐴𝐴 𝐴 ∈ V)
1918con3i 150 . 2 𝐴 ∈ V → ¬ 𝒫 𝒫 𝐴𝐴)
2014, 19pm2.61i 176 1 ¬ 𝒫 𝒫 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2030  Vcvv 3231  wss 3607  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  cdom 7995  csdm 7996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000
This theorem is referenced by:  mnfnre  10120
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