Theorem bdcpw 12026

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 12021	. . 3 |- BOUNDED x C_ A
3	2	bdcab 12006	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3435	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 12001	1 |- BOUNDED ~P A

Syntax hints:   {cab 2075   C_ wss 3000   ~Pcpw 3433  BOUNDED wbdc 11997

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-bd0 11970  ax-bdal 11975  ax-bdsb 11979

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-ral 2365  df-in 3006  df-ss 3013  df-pw 3435  df-bdc 11998

This theorem is referenced by: (None)