Theorem bdcpw 15361

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 15356	. . 3 |- BOUNDED x C_ A
3	2	bdcab 15341	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3603	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 15336	1 |- BOUNDED ~P A

Syntax hints:   {cab 2179   C_ wss 3153   ~Pcpw 3601  BOUNDED wbdc 15332

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15305  ax-bdal 15310  ax-bdsb 15314

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-in 3159  df-ss 3166  df-pw 3603  df-bdc 15333

This theorem is referenced by: (None)