Theorem pwssunim 4319

Description: The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.)

Assertion

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

Proof of Theorem pwssunim

Step	Hyp	Ref	Expression
1		ssequn2 3336	. . . . 5 |- ( B C_ A <-> ( A u. B ) = A )
2		pweq 3608	. . . . . 6 |- ( ( A u. B ) = A -> ~P ( A u. B ) = ~P A )
3		eqimss 3237	. . . . . 6 |- ( ~P ( A u. B ) = ~P A -> ~P ( A u. B ) C_ ~P A )
4	2, 3	syl 14	. . . . 5 |- ( ( A u. B ) = A -> ~P ( A u. B ) C_ ~P A )
5	1, 4	sylbi 121	. . . 4 |- ( B C_ A -> ~P ( A u. B ) C_ ~P A )
6		ssequn1 3333	. . . . 5 |- ( A C_ B <-> ( A u. B ) = B )
7		pweq 3608	. . . . . 6 |- ( ( A u. B ) = B -> ~P ( A u. B ) = ~P B )
8		eqimss 3237	. . . . . 6 |- ( ~P ( A u. B ) = ~P B -> ~P ( A u. B ) C_ ~P B )
9	7, 8	syl 14	. . . . 5 |- ( ( A u. B ) = B -> ~P ( A u. B ) C_ ~P B )
10	6, 9	sylbi 121	. . . 4 |- ( A C_ B -> ~P ( A u. B ) C_ ~P B )
11	5, 10	orim12i 760	. . 3 |- ( ( B C_ A \/ A C_ B ) -> ( ~P ( A u. B ) C_ ~P A \/ ~P ( A u. B ) C_ ~P B ) )
12	11	orcoms 731	. 2 |- ( ( A C_ B \/ B C_ A ) -> ( ~P ( A u. B ) C_ ~P A \/ ~P ( A u. B ) C_ ~P B ) )
13		ssun 3342	. 2 |- ( ( ~P ( A u. B ) C_ ~P A \/ ~P ( A u. B ) C_ ~P B ) -> ~P ( A u. B ) C_ ( ~P A u. ~P B ) )
14	12, 13	syl 14	1 |- ( ( A C_ B \/ B C_ A ) -> ~P ( A u. B ) C_ ( ~P A u. ~P B ) )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   u. cun 3155   C_ wss 3157   ~Pcpw 3605

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607

This theorem is referenced by:  pwunim  4321