Theorem bdcpw 13711

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 13706	. . 3 |- BOUNDED x C_ A
3	2	bdcab 13691	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3560	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 13686	1 |- BOUNDED ~P A

Syntax hints:   {cab 2151   C_ wss 3115   ~Pcpw 3558  BOUNDED wbdc 13682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bd0 13655  ax-bdal 13660  ax-bdsb 13664

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2448  df-in 3121  df-ss 3128  df-pw 3560  df-bdc 13683

This theorem is referenced by: (None)