Theorem elpwpwel 4477

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 4475	. . 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 3975	. 2 |- ( A e. ~P ~P B <-> ( A e. _V /\ U. A C_ B ) )
4		elpwb 3587	. 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 2148   _Vcvv 2739   C_ wss 3131   ~Pcpw 3577   U.cuni 3811

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579  df-uni 3812

This theorem is referenced by: (None)