Theorem pwun 5022

Description: The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by NM, 23-Nov-2003.)

Assertion

Ref	Expression
pwun	|- ( ( A C_ B \/ B C_ A ) <-> ~P ( A u. B ) = ( ~P A u. ~P B ) )

Proof of Theorem pwun

Step	Hyp	Ref	Expression
1		pwunss 5019	. . 3 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
2	1	biantru 526	. 2 |- ( ~P ( A u. B ) C_ ( ~P A u. ~P B ) <-> ( ~P ( A u. B ) C_ ( ~P A u. ~P B ) /\ ( ~P A u. ~P B ) C_ ~P ( A u. B ) ) )
3		pwssun 5020	. 2 |- ( ( A C_ B \/ B C_ A ) <-> ~P ( A u. B ) C_ ( ~P A u. ~P B ) )
4		eqss 3618	. 2 |- ( ~P ( A u. B ) = ( ~P A u. ~P B ) <-> ( ~P ( A u. B ) C_ ( ~P A u. ~P B ) /\ ( ~P A u. ~P B ) C_ ~P ( A u. B ) ) )
5	2, 3, 4	3bitr4i 292	1 |- ( ( A C_ B \/ B C_ A ) <-> ~P ( A u. B ) = ( ~P A u. ~P B ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   u. cun 3572   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  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-un 3579  df-in 3581  df-ss 3588  df-pw 4160  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)