Theorem pwne 4243

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3886. (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 4242	. 2 |- ( A e. V -> -. ~P A C_ A )
2		eqimss 3278	. . 3 |- ( ~P A = A -> ~P A C_ A )
3	2	necon3bi 2450	. 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 2200   =/= wne 2400   C_ wss 3197   ~Pcpw 3649

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4201

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by:  pnfnemnf  8189