Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdcpw GIF version

Theorem bdcpw 12994
Description: The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
Hypothesis
Ref Expression
bdcpw.1 BOUNDED 𝐴
Assertion
Ref Expression
bdcpw BOUNDED 𝒫 𝐴

Proof of Theorem bdcpw
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 bdcpw.1 . . . 4 BOUNDED 𝐴
21bdss 12989 . . 3 BOUNDED 𝑥𝐴
32bdcab 12974 . 2 BOUNDED {𝑥𝑥𝐴}
4 df-pw 3482 . 2 𝒫 𝐴 = {𝑥𝑥𝐴}
53, 4bdceqir 12969 1 BOUNDED 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  {cab 2103  wss 3041  𝒫 cpw 3480  BOUNDED wbdc 12965
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-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bd0 12938  ax-bdal 12943  ax-bdsb 12947
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-in 3047  df-ss 3054  df-pw 3482  df-bdc 12966
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator