Theorem pwunim 4271

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 Jim Kingdon, 30-Sep-2018.)

Assertion

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

Proof of Theorem pwunim

Step	Hyp	Ref	Expression
1		pwssunim 4269	. . 3 |- ( ( A C_ B \/ B C_ A ) -> ~P ( A u. B ) C_ ( ~P A u. ~P B ) )
2		pwunss 4268	. . . 4 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3	2	biantru 300	. . 3 |- ( ~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 ) ) )
4	1, 3	sylib 121	. 2 |- ( ( A C_ B \/ B C_ A ) -> ( ~P ( A u. B ) C_ ( ~P A u. ~P B ) /\ ( ~P A u. ~P B ) C_ ~P ( A u. B ) ) )
5		eqss 3162	. 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 ) ) )
6	4, 5	sylibr 133	1 |- ( ( A C_ B \/ B C_ A ) -> ~P ( A u. B ) = ( ~P A u. ~P B ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   u. cun 3119   C_ wss 3121   ~Pcpw 3566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by: (None)