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Theorem 2pwuninelg 5980
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 Jim Kingdon, 14-Jan-2020.)
Assertion
Ref Expression
2pwuninelg (𝐴𝑉 → ¬ 𝒫 𝒫 𝐴𝐴)

Proof of Theorem 2pwuninelg
StepHypRef Expression
1 en2lp 4333 . 2 ¬ (𝐴 ∈ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴𝐴)
2 pwuni 3991 . . . 4 𝐴 ⊆ 𝒫 𝐴
3 elpwg 3414 . . . 4 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐴𝐴 ⊆ 𝒫 𝐴))
42, 3mpbiri 166 . . 3 (𝐴𝑉𝐴 ∈ 𝒫 𝒫 𝐴)
5 ax-ia3 106 . . 3 (𝐴 ∈ 𝒫 𝒫 𝐴 → (𝒫 𝒫 𝐴𝐴 → (𝐴 ∈ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴𝐴)))
64, 5syl 14 . 2 (𝐴𝑉 → (𝒫 𝒫 𝐴𝐴 → (𝐴 ∈ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴𝐴)))
71, 6mtoi 623 1 (𝐴𝑉 → ¬ 𝒫 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wcel 1434  wss 2984  𝒫 cpw 3406   cuni 3627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-setind 4316
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628
This theorem is referenced by:  mnfnre  7433
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