Theorem elpwi2 4242

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 2812	. . 3 |- B e. _V
4	3	elpw2 4241	. 2 |- ( A e. ~P B <-> A C_ B )
5	1, 4	mpbir 146	1 |- A e. ~P B

Syntax hints:   e. wcel 2200   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-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 4202

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

This theorem is referenced by:  canth  5952  bitsf  12457  prdsvallem  13305