Theorem bdcpw 15515

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 15510	. . 3 |- BOUNDED x C_ A
3	2	bdcab 15495	. 2 |- BOUNDED { x | x C_ A }
4		df-pw 3607	. 2 |- ~P A = { x | x C_ A }
5	3, 4	bdceqir 15490	1 |- BOUNDED ~P A

Syntax hints:   {cab 2182   C_ wss 3157   ~Pcpw 3605  BOUNDED wbdc 15486

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15459  ax-bdal 15464  ax-bdsb 15468

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-in 3163  df-ss 3170  df-pw 3607  df-bdc 15487

This theorem is referenced by: (None)