Theorem elpwd 4167

Description: Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

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

Assertion

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

Proof of Theorem elpwd

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

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   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

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:  reprval  30688  scutval  31911  bj-discrmoore  33066  dmvolss  40202  sge0xaddlem1  40650  ovnval2b  40766  ovnsubadd2lem  40859  vonvolmbllem  40874  vonvolmbl  40875  smfresal  40995  smfpimbor1lem1  41005  sprsymrelfvlem  41740