Theorem snelpwrVD 39066

Description: Virtual deduction proof of snelpwi 4912. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
snelpwrVD	|- ( A e. B -> { A } e. ~P B )

Proof of Theorem snelpwrVD

Step	Hyp	Ref	Expression
1		snex 4908	. . 3 |- { A } e. _V
2		idn1 38790	. . . 4 |- (. A e. B ->. A e. B ).
3		snssi 4339	. . . 4 |- ( A e. B -> { A } C_ B )
4	2, 3	e1a 38852	. . 3 |- (. A e. B ->. { A } C_ B ).
5		elpwg 4166	. . . 4 |- ( { A } e. _V -> ( { A } e. ~P B <-> { A } C_ B ) )
6	5	biimprd 238	. . 3 |- ( { A } e. _V -> ( { A } C_ B -> { A } e. ~P B ) )
7	1, 4, 6	e01 38916	. 2 |- (. A e. B ->. { A } e. ~P B ).
8	7	in1 38787	1 |- ( A e. B -> { A } e. ~P B )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177

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  ax-nul 4789  ax-pr 4906

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-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-vd1 38786

This theorem is referenced by: (None)