Theorem bdcpw 16315

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 16310	. . 3 |- BOUNDED x C_ A
3	2	bdcab 16295	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3651	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 16290	1 |- BOUNDED ~P A

Syntax hints:   {cab 2215   C_ wss 3197   ~Pcpw 3649  BOUNDED wbdc 16286

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16259  ax-bdal 16264  ax-bdsb 16268

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-in 3203  df-ss 3210  df-pw 3651  df-bdc 16287

This theorem is referenced by: (None)