Theorem elpwgded 38780

Description: elpwgdedVD 39153 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elpwgded.1	|- ( ph -> A e. _V )
elpwgded.2	|- ( ps -> A C_ B )

Assertion

Ref	Expression
elpwgded	|- ( ( ph /\ ps ) -> A e. ~P B )

Proof of Theorem elpwgded

Step	Hyp	Ref	Expression
1		elpwgded.1	. 2 |- ( ph -> A e. _V )
2		elpwgded.2	. 2 |- ( ps -> A C_ B )
3		elpwg 4166	. . 3 |- ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
4	3	biimpar 502	. 2 |- ( ( A e. _V /\ A C_ B ) -> A e. ~P B )
5	1, 2, 4	syl2an 494	1 |- ( ( ph /\ ps ) -> A e. ~P B )

Syntax hints:   -> wi 4   /\ wa 384   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

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:  sspwimp  39154  sspwimpALT  39161