Theorem elpwdifcl 29358

Description: Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)

Hypothesis

Ref	Expression
elpwincl.1	|- ( ph -> A e. ~P C )

Assertion

Ref	Expression
elpwdifcl	|- ( ph -> ( A \ B ) e. ~P C )

Proof of Theorem elpwdifcl

Step	Hyp	Ref	Expression
1		elpwincl.1	. . . 4 |- ( ph -> A e. ~P C )
2	1	elpwid 4170	. . 3 |- ( ph -> A C_ C )
3	2	ssdifssd 3748	. 2 |- ( ph -> ( A \ B ) C_ C )
4		difexg 4808	. . 3 |- ( A e. ~P C -> ( A \ B ) e. _V )
5		elpwg 4166	. . 3 |- ( ( A \ B ) e. _V -> ( ( A \ B ) e. ~P C <-> ( A \ B ) C_ C ) )
6	1, 4, 5	3syl 18	. 2 |- ( ph -> ( ( A \ B ) e. ~P C <-> ( A \ B ) C_ C ) )
7	3, 6	mpbird 247	1 |- ( ph -> ( A \ B ) e. ~P C )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  pwldsys  30220  ldgenpisyslem1  30226  difelcarsg  30372  inelcarsg  30373  carsgclctunlem2  30381  carsgclctunlem3  30382  carsgclctun  30383