Theorem bdcpw 14974

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 14969	. . 3 ⊢ BOUNDED 𝑥 ⊆ 𝐴
3	2	bdcab 14954	. 2 ⊢ BOUNDED {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 3589	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
5	3, 4	bdceqir 14949	1 ⊢ BOUNDED 𝒫 𝐴

Syntax hints:  {cab 2173   ⊆ wss 3141  𝒫 cpw 3587  BOUNDED wbdc 14945

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-bd0 14918  ax-bdal 14923  ax-bdsb 14927

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-ral 2470  df-in 3147  df-ss 3154  df-pw 3589  df-bdc 14946

This theorem is referenced by: (None)