Theorem bdcpw 10818

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 10813	. . 3 ⊢ BOUNDED 𝑥 ⊆ 𝐴
3	2	bdcab 10798	. 2 ⊢ BOUNDED {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 3392	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
5	3, 4	bdceqir 10793	1 ⊢ BOUNDED 𝒫 𝐴

Syntax hints:  {cab 2068   ⊆ wss 2974  𝒫 cpw 3390  BOUNDED wbdc 10789

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-bd0 10762  ax-bdal 10767  ax-bdsb 10771

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-in 2980  df-ss 2987  df-pw 3392  df-bdc 10790

This theorem is referenced by: (None)