Theorem bdcpw 14181

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 14176	. . 3 ⊢ BOUNDED 𝑥 ⊆ 𝐴
3	2	bdcab 14161	. 2 ⊢ BOUNDED {𝑥 ∣ 𝑥 ⊆ 𝐴}
4		df-pw 3574	. 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
5	3, 4	bdceqir 14156	1 ⊢ BOUNDED 𝒫 𝐴

Syntax hints:  {cab 2161   ⊆ wss 3127  𝒫 cpw 3572  BOUNDED wbdc 14152

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-bd0 14125  ax-bdal 14130  ax-bdsb 14134

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-ral 2458  df-in 3133  df-ss 3140  df-pw 3574  df-bdc 14153

This theorem is referenced by: (None)