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Theorem pwv 3806
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 3177 . . . 4 𝑥 ⊆ V
2 vex 2740 . . . . 5 𝑥 ∈ V
32elpw 3580 . . . 4 (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V)
41, 3mpbir 146 . . 3 𝑥 ∈ 𝒫 V
54, 22th 174 . 2 (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V)
65eqriv 2174 1 𝒫 V = V
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2148  Vcvv 2737  wss 3129  𝒫 cpw 3574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576
This theorem is referenced by:  univ  4472
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