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Theorem pwv 3705
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 3089 . . . 4 𝑥 ⊆ V
2 vex 2663 . . . . 5 𝑥 ∈ V
32elpw 3486 . . . 4 (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
41, 3mpbir 145 . . 3 𝑥 ∈ 𝒫 V
54, 22th 173 . 2 (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
65eqriv 2114 1 𝒫 V = V
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wcel 1465  Vcvv 2660  wss 3041  𝒫 cpw 3480
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482
This theorem is referenced by:  univ  4367
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