Theorem elpwunicl 29371

Description: Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Hypotheses

Ref	Expression
elpwunicl.1	|- ( ph -> B e. V )
elpwunicl.2	|- ( ph -> A e. ~P ~P B )

Assertion

Ref	Expression
elpwunicl	|- ( ph -> U. A e. ~P B )

Proof of Theorem elpwunicl

Step	Hyp	Ref	Expression
1		elpwunicl.2	. . . 4 |- ( ph -> A e. ~P ~P B )
2		elpwg 4166	. . . . 5 |- ( A e. ~P ~P B -> ( A e. ~P ~P B <-> A C_ ~P B ) )
3	1, 2	syl 17	. . . 4 |- ( ph -> ( A e. ~P ~P B <-> A C_ ~P B ) )
4	1, 3	mpbid 222	. . 3 |- ( ph -> A C_ ~P B )
5		pwuniss 29370	. . 3 |- ( A C_ ~P B -> U. A C_ B )
6	4, 5	syl 17	. 2 |- ( ph -> U. A C_ B )
7		uniexg 6955	. . 3 |- ( A e. ~P ~P B -> U. A e. _V )
8		elpwg 4166	. . 3 |- ( U. A e. _V -> ( U. A e. ~P B <-> U. A C_ B ) )
9	1, 7, 8	3syl 18	. 2 |- ( ph -> ( U. A e. ~P B <-> U. A C_ B ) )
10	6, 9	mpbird 247	1 |- ( ph -> U. A e. ~P B )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  ldgenpisyslem1  30226