Theorem pwne 4256

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3897. (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 4255	. 2 |- ( A e. V -> -. ~P A C_ A )
2		eqimss 3282	. . 3 |- ( ~P A = A -> ~P A C_ A )
3	2	necon3bi 2453	. 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 2202   =/= wne 2403   C_ wss 3201   ~Pcpw 3656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-rab 2520  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658

This theorem is referenced by:  pnfnemnf  8277