Theorem bdcpw 16004

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 15999	. . 3 ⊢ BOUNDED 𝑥 ⊆ 𝐴
3	2	bdcab 15984	. 2 ⊢ BOUNDED {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 3628	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
5	3, 4	bdceqir 15979	1 ⊢ BOUNDED 𝒫 𝐴

Syntax hints:  {cab 2193   ⊆ wss 3174  𝒫 cpw 3626  BOUNDED wbdc 15975

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-bd0 15948  ax-bdal 15953  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-in 3180  df-ss 3187  df-pw 3628  df-bdc 15976

This theorem is referenced by: (None)