Theorem pwne 4208

Description: No set equals its power set. The sethood antecedent is necessary; compare pwv 3851. (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 4207	. 2 |- ( A e. V -> -. ~P A C_ A )
2		eqimss 3248	. . 3 |- ( ~P A = A -> ~P A C_ A )
3	2	necon3bi 2427	. 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 2177   =/= wne 2377   C_ wss 3167   ~Pcpw 3617

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-sep 4166

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-rab 2494  df-v 2775  df-in 3173  df-ss 3180  df-pw 3619

This theorem is referenced by:  pnfnemnf  8134