Theorem bdcpw 14503

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 14498	. . 3 |- BOUNDED x C_ A
3	2	bdcab 14483	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3577	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 14478	1 |- BOUNDED ~P A

Syntax hints:   {cab 2163   C_ wss 3129   ~Pcpw 3575  BOUNDED wbdc 14474

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14447  ax-bdal 14452  ax-bdsb 14456

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-in 3135  df-ss 3142  df-pw 3577  df-bdc 14475

This theorem is referenced by: (None)