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Theorem 2pwuninelg 5929
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 4306 . 2 ¬ (𝐴 ∈ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴𝐴)
2 pwuni 3971 . . . 4 𝐴 ⊆ 𝒫 𝐴
3 elpwg 3395 . . . 4 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐴𝐴 ⊆ 𝒫 𝐴))
42, 3mpbiri 161 . . 3 (𝐴𝑉𝐴 ∈ 𝒫 𝒫 𝐴)
5 ax-ia3 105 . . 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 600 1 (𝐴𝑉 → ¬ 𝒫 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wcel 1409  wss 2945  𝒫 cpw 3387   cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609
This theorem is referenced by:  mnfnre  7127
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