Theorem bdcpw 10660

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.

Step	Hyp	Ref	Expression
1		bdcpw.1	. . . 4 |- BOUNDED A
2	1	bdss 10655	. . 3 |- BOUNDED x C_ A
3	2	bdcab 10640	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3384	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 10635	1 |- BOUNDED ~P A

Syntax hints:   {cab 2067   C_ wss 2973   ~Pcpw 3382  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdal 10609  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-in 2979  df-ss 2986  df-pw 3384  df-bdc 10632

This theorem is referenced by: (None)