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Theorem pwv 3549
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 2938 . . . 4 x ⊆ V
2 vex 2534 . . . . 5 x V
32elpw 3336 . . . 4 (x 𝒫 V ↔ x ⊆ V)
41, 3mpbir 134 . . 3 x 𝒫 V
54, 22th 163 . 2 (x 𝒫 V ↔ x V)
65eqriv 2015 1 𝒫 V = V
Colors of variables: wff set class
Syntax hints:   = wceq 1226   wcel 1370  Vcvv 2531  wss 2890  𝒫 cpw 3330
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-in 2897  df-ss 2904  df-pw 3332
This theorem is referenced by:  univ  4153
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