Theorem elpwi2 4218

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 2789	. . 3 |- B e. _V
4	3	elpw2 4217	. 2 |- ( A e. ~P B <-> A C_ B )
5	1, 4	mpbir 146	1 |- A e. ~P B

Syntax hints:   e. wcel 2178   C_ wss 3174   ~Pcpw 3626

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 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 2189  ax-sep 4178

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628

This theorem is referenced by:  canth  5920  bitsf  12372  prdsvallem  13219