Theorem bdcpw 13238

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 13233	. . 3 |- BOUNDED x C_ A
3	2	bdcab 13218	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3517	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 13213	1 |- BOUNDED ~P A

Syntax hints:   {cab 2126   C_ wss 3076   ~Pcpw 3515  BOUNDED wbdc 13209

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bd0 13182  ax-bdal 13187  ax-bdsb 13191

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-in 3082  df-ss 3089  df-pw 3517  df-bdc 13210

This theorem is referenced by: (None)