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Theorem 2pwuninelg 6148
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 4439 . 2 ¬ (𝐴 ∈ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴𝐴)
2 pwuni 4086 . . . 4 𝐴 ⊆ 𝒫 𝐴
3 elpwg 3488 . . . 4 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐴𝐴 ⊆ 𝒫 𝐴))
42, 3mpbiri 167 . . 3 (𝐴𝑉𝐴 ∈ 𝒫 𝒫 𝐴)
5 ax-ia3 107 . . 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 638 1 (𝐴𝑉 → ¬ 𝒫 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wcel 1465  wss 3041  𝒫 cpw 3480   cuni 3706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707
This theorem is referenced by:  mnfnre  7776
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