Theorem pwunim 4041

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 4039	. . 3 |- ( ( A C_ B \/ B C_ A ) -> ~P ( A u. B ) C_ ( ~P A u. ~P B ) )
2		pwunss 4038	. . . 4 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3	2	biantru 296	. . 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 120	. 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 3014	. 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 132	1 |- ( ( A C_ B \/ B C_ A ) -> ~P ( A u. B ) = ( ~P A u. ~P B ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   = wceq 1284   u. cun 2971   C_ wss 2973   ~Pcpw 3382

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by: (None)