Theorem pwne 4831

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 4433. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
pwne	|- ( A e. V -> ~P A =/= A )

Proof of Theorem pwne

Step	Hyp	Ref	Expression
1		pwnss 4830	. 2 |- ( A e. V -> -. ~P A C_ A )
2		eqimss 3657	. . 3 |- ( ~P A = A -> ~P A C_ A )
3	2	necon3bi 2820	. 2 |- ( -. ~P A C_ A -> ~P A =/= A )
4	1, 3	syl 17	1 |- ( A e. V -> ~P A =/= A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   =/= wne 2794   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-ne 2795  df-nel 2898  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  pnfnemnf  10094