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Theorem 2pwuninelg 6086
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 4398 . 2 ¬ (𝐴 ∈ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴𝐴)
2 pwuni 4048 . . . 4 𝐴 ⊆ 𝒫 𝐴
3 elpwg 3457 . . . 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 628 1 (𝐴𝑉 → ¬ 𝒫 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wcel 1445  wss 3013  𝒫 cpw 3449   cuni 3675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676
This theorem is referenced by:  mnfnre  7627
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