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Theorem pwv 3742
 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 3123 . . . 4 𝑥 ⊆ V
2 vex 2692 . . . . 5 𝑥 ∈ V
32elpw 3520 . . . 4 (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
41, 3mpbir 145 . . 3 𝑥 ∈ 𝒫 V
54, 22th 173 . 2 (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
65eqriv 2137 1 𝒫 V = V
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∈ wcel 1481  Vcvv 2689   ⊆ wss 3075  𝒫 cpw 3514 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3081  df-ss 3088  df-pw 3516 This theorem is referenced by:  univ  4404
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