Theorem elpwi2 4203

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 2784	. . 3 |- B e. _V
4	3	elpw2 4202	. 2 |- ( A e. ~P B <-> A C_ B )
5	1, 4	mpbir 146	1 |- A e. ~P B

Syntax hints:   e. wcel 2176   C_ wss 3166   ~Pcpw 3616

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-sep 4163

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618

This theorem is referenced by:  canth  5899  bitsf  12290  prdsvallem  13137