Theorem bdcpw 16464

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 16459	. . 3 |- BOUNDED x C_ A
3	2	bdcab 16444	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3654	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 16439	1 |- BOUNDED ~P A

Syntax hints:   {cab 2217   C_ wss 3200   ~Pcpw 3652  BOUNDED wbdc 16435

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-bd0 16408  ax-bdal 16413  ax-bdsb 16417

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-in 3206  df-ss 3213  df-pw 3654  df-bdc 16436

This theorem is referenced by: (None)