Theorem bdcpw 16639

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 16634	. . 3 |- BOUNDED x C_ A
3	2	bdcab 16619	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3671	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 16614	1 |- BOUNDED ~P A

Syntax hints:   {cab 2218   C_ wss 3211   ~Pcpw 3669  BOUNDED wbdc 16610

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-bd0 16583  ax-bdal 16588  ax-bdsb 16592

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-ral 2525  df-in 3217  df-ss 3224  df-pw 3671  df-bdc 16611

This theorem is referenced by: (None)