Theorem pwne 4295

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3932. (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 4294	. 2 |- ( A e. V -> -. ~P A C_ A )
2		eqimss 3302	. . 3 |- ( ~P A = A -> ~P A C_ A )
3	2	necon3bi 2470	. 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 2209   =/= wne 2420   C_ wss 3220   ~Pcpw 3688

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4247

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-rab 2537  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690

This theorem is referenced by:  pnfnemnf  8374