Theorem pwne 4146

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3795. (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 4145	. 2 |- ( A e. V -> -. ~P A C_ A )
2		eqimss 3201	. . 3 |- ( ~P A = A -> ~P A C_ A )
3	2	necon3bi 2390	. 2 |- ( -. ~P A C_ A -> ~P A =/= A )
4	1, 3	syl 14	1 |- ( A e. V -> ~P A =/= A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2141   =/= wne 2340   C_ wss 3121   ~Pcpw 3566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  pnfnemnf  7974