Theorem bdcpw 13056

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 13051	. . 3 |- BOUNDED x C_ A
3	2	bdcab 13036	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3507	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 13031	1 |- BOUNDED ~P A

Syntax hints:   {cab 2123   C_ wss 3066   ~Pcpw 3505  BOUNDED wbdc 13027

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-bd0 13000  ax-bdal 13005  ax-bdsb 13009

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-in 3072  df-ss 3079  df-pw 3507  df-bdc 13028

This theorem is referenced by: (None)