Theorem bdcpw 13904

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 13899	. . 3 |- BOUNDED x C_ A
3	2	bdcab 13884	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3568	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 13879	1 |- BOUNDED ~P A

Syntax hints:   {cab 2156   C_ wss 3121   ~Pcpw 3566  BOUNDED wbdc 13875

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bd0 13848  ax-bdal 13853  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-in 3127  df-ss 3134  df-pw 3568  df-bdc 13876

This theorem is referenced by: (None)