Theorem sselpwd 4807

Description: Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)

Hypotheses

Ref	Expression
sselpwd.1	|- ( ph -> B e. V )
sselpwd.2	|- ( ph -> A C_ B )

Assertion

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

Proof of Theorem sselpwd

Step	Hyp	Ref	Expression
1		sselpwd.2	. 2 |- ( ph -> A C_ B )
2		sselpwd.1	. . . 4 |- ( ph -> B e. V )
3	2, 1	ssexd 4805	. . 3 |- ( ph -> A e. _V )
4		elpwg 4166	. . 3 |- ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
5	3, 4	syl 17	. 2 |- ( ph -> ( A e. ~P B <-> A C_ B ) )
6	1, 5	mpbird 247	1 |- ( ph -> A e. ~P B )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   _Vcvv 3200   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-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  knatar  6607  fin1a2lem7  9228  wunss  9534  mreexd  16302  mreexexlemd  16304  ustssel  22009  crefi  29914  ldsysgenld  30223  ldgenpisyslem1  30226  rfovcnvf1od  38298  fsovrfovd  38303  fsovfd  38306  fsovcnvlem  38307  ntrclsrcomplex  38333  clsk3nimkb  38338  clsk1indlem3  38341  clsk1indlem4  38342  clsk1indlem1  38343  ntrclsiso  38365  ntrclskb  38367  ntrclsk3  38368  ntrclsk13  38369  ntrneircomplex  38372  ntrneik3  38394  ntrneix3  38395  ntrneik13  38396  ntrneix13  38397  clsneircomplex  38401  clsneiel1  38406  neicvgrcomplex  38411  neicvgel1  38417  ovolsplit  40205