Theorem elpwpwel 4507

Description: A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)

Assertion

Ref	Expression
elpwpwel	|- ( A e. ~P ~P B <-> U. A e. ~P B )

Proof of Theorem elpwpwel

Step	Hyp	Ref	Expression
1		uniexb 4505	. . 3 |- ( A e. _V <-> U. A e. _V )
2	1	anbi1i 458	. 2 |- ( ( A e. _V /\ U. A C_ B ) <-> ( U. A e. _V /\ U. A C_ B ) )
3		elpwpw 4000	. 2 |- ( A e. ~P ~P B <-> ( A e. _V /\ U. A C_ B ) )
4		elpwb 3612	. 2 |- ( U. A e. ~P B <-> ( U. A e. _V /\ U. A C_ B ) )
5	2, 3, 4	3bitr4i 212	1 |- ( A e. ~P ~P B <-> U. A e. ~P B )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2164   _Vcvv 2760   C_ wss 3154   ~Pcpw 3602   U.cuni 3836

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3160  df-ss 3167  df-pw 3604  df-uni 3837

This theorem is referenced by: (None)