Theorem elpwi2 4289

Description: Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)

Hypotheses

Ref	Expression
elpwi2.1	|- B e. V
elpwi2.2	|- A C_ B

Assertion

Ref	Expression
elpwi2	|- A e. ~P B

Proof of Theorem elpwi2

Step	Hyp	Ref	Expression
1		elpwi2.2	. 2 |- A C_ B
2		elpwi2.1	. . . 4 |- B e. V
3	2	elexi 2834	. . 3 |- B e. _V
4	3	elpw2 4288	. 2 |- ( A e. ~P B <-> A C_ B )
5	1, 4	mpbir 146	1 |- A e. ~P B

Syntax hints:   e. wcel 2209   C_ wss 3220   ~Pcpw 3685

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-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 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by:  canth  6026  bitsf  12691  ballotfilem2  13206  ballotfilemth  13259  prdsvallem  13598