Theorem pwunim 4315

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 4313	. . 3 |- ( ( A C_ B \/ B C_ A ) -> ~P ( A u. B ) C_ ( ~P A u. ~P B ) )
2		pwunss 4312	. . . 4 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3	2	biantru 302	. . 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 122	. 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 3194	. 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 134	1 |- ( ( A C_ B \/ B C_ A ) -> ~P ( A u. B ) = ( ~P A u. ~P B ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   u. cun 3151   C_ wss 3153   ~Pcpw 3601

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603

This theorem is referenced by: (None)