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Theorem 2pwuninel 8059
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 8041 . . 3 ¬ 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴
2 elssuni 4433 . . . 4 (𝒫 𝒫 𝐴𝐴 → 𝒫 𝒫 𝐴 𝐴)
3 ssdomg 7945 . . . . 5 ( 𝐴 ∈ V → (𝒫 𝒫 𝐴 𝐴 → 𝒫 𝒫 𝐴 𝐴))
4 canth2g 8058 . . . . . 6 ( 𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
5 pwexb 6922 . . . . . . 7 ( 𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
6 canth2g 8058 . . . . . . 7 (𝒫 𝐴 ∈ V → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴)
75, 6sylbi 207 . . . . . 6 ( 𝐴 ∈ V → 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴)
8 sdomtr 8042 . . . . . 6 (( 𝐴 ≺ 𝒫 𝐴 ∧ 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴) → 𝐴 ≺ 𝒫 𝒫 𝐴)
94, 7, 8syl2anc 692 . . . . 5 ( 𝐴 ∈ V → 𝐴 ≺ 𝒫 𝒫 𝐴)
10 domsdomtr 8039 . . . . . 6 ((𝒫 𝒫 𝐴 𝐴 𝐴 ≺ 𝒫 𝒫 𝐴) → 𝒫 𝒫 𝐴 ≺ 𝒫 𝒫 𝐴)
1110ex 450 . . . . 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 3198 . . . 4 (𝒫 𝒫 𝐴𝐴 → 𝒫 𝒫 𝐴 ∈ V)
16 pwexb 6922 . . . . 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 1987  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  cdom 7897  csdm 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902
This theorem is referenced by:  mnfnre  10026
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