ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwnex GIF version

Theorem pwnex 4373
Description: The class of all power sets is a proper class. See also snnex 4372. (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 4371 . . 3 (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
2 df-nel 2404 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∈ V)
31, 2sylibr 133 . 2 (∀𝑦(𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦) → {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V)
4 vpwex 4106 . . 3 𝒫 𝑦 ∈ V
5 vex 2689 . . . 4 𝑦 ∈ V
65pwid 3525 . . 3 𝑦 ∈ 𝒫 𝑦
74, 6pm3.2i 270 . 2 (𝒫 𝑦 ∈ V ∧ 𝑦 ∈ 𝒫 𝑦)
83, 7mpg 1427 1 {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wal 1329   = wceq 1331  wex 1468  wcel 1480  {cab 2125  wnel 2403  Vcvv 2686  𝒫 cpw 3510
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-un 4358
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-nel 2404  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-uni 3740  df-iun 3818
This theorem is referenced by:  topnex  12281
  Copyright terms: Public domain W3C validator