Theorem elpwi2 4829

Description: Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

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	. . 3 |- B e. V
3		elpw2g 4827	. . 3 |- ( B e. V -> ( A e. ~P B <-> A C_ B ) )
4	2, 3	ax-mp 5	. 2 |- ( A e. ~P B <-> A C_ B )
5	1, 4	mpbir 221	1 |- A e. ~P B

Syntax hints:   <-> wb 196   e. wcel 1990   C_ wss 3574   ~Pcpw 4158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  sprsymrelfolem1  41742