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Theorem pwnex 4470
Description: The class of all power sets is a proper class. See also snnex 4469. (Contributed by BJ, 2-May-2021.)
Assertion
Ref Expression
pwnex {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V
Distinct variable group:   𝑥,𝑦

Proof of Theorem pwnex
StepHypRef Expression
1 abnex 4468 . . 3 (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
2 df-nel 2456 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
31, 2sylibr 134 . 2 (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V)
4 vpwex 4200 . . 3 𝒫 𝑦 ∈ V
5 vex 2755 . . . 4 𝑦 ∈ V
65pwid 3608 . . 3 𝑦 ∈ 𝒫 𝑦
74, 6pm3.2i 272 . 2 (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
83, 7mpg 1462 1 {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wal 1362   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wnel 2455  Vcvv 2752  𝒫 cpw 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-un 4454
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-nel 2456  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-uni 3828  df-iun 3906
This theorem is referenced by:  topnex  14071
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