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Theorem pwv 3570
Description: The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
Assertion
Ref Expression
pwv 𝒫 V = V

Proof of Theorem pwv
StepHypRef Expression
1 ssv 2959 . . . 4 x ⊆ V
2 vex 2554 . . . . 5 x V
32elpw 3357 . . . 4 (x 𝒫 V ↔ x ⊆ V)
41, 3mpbir 134 . . 3 x 𝒫 V
54, 22th 163 . 2 (x 𝒫 V ↔ x V)
65eqriv 2034 1 𝒫 V = V
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  wss 2911  𝒫 cpw 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353
This theorem is referenced by:  univ  4173
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