Theorem bdcpw 16190

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 16185	. . 3 ⊢ BOUNDED 𝑥 ⊆ 𝐴
3	2	bdcab 16170	. 2 ⊢ BOUNDED {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 3651	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
5	3, 4	bdceqir 16165	1 ⊢ BOUNDED 𝒫 𝐴

Syntax hints:  {cab 2215   ⊆ wss 3197  𝒫 cpw 3649  BOUNDED wbdc 16161

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16134  ax-bdal 16139  ax-bdsb 16143

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-in 3203  df-ss 3210  df-pw 3651  df-bdc 16162

This theorem is referenced by: (None)