Theorem pwssunim 4381

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 3380	. . . . 5 |- ( B C_ A <-> ( A u. B ) = A )
2		pweq 3655	. . . . . 6 |- ( ( A u. B ) = A -> ~P ( A u. B ) = ~P A )
3		eqimss 3281	. . . . . 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 3377	. . . . 5 |- ( A C_ B <-> ( A u. B ) = B )
7		pweq 3655	. . . . . 6 |- ( ( A u. B ) = B -> ~P ( A u. B ) = ~P B )
8		eqimss 3281	. . . . . 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 766	. . 3 |- ( ( B C_ A \/ A C_ B ) -> ( ~P ( A u. B ) C_ ~P A \/ ~P ( A u. B ) C_ ~P B ) )
12	11	orcoms 737	. 2 |- ( ( A C_ B \/ B C_ A ) -> ( ~P ( A u. B ) C_ ~P A \/ ~P ( A u. B ) C_ ~P B ) )
13		ssun 3386	. 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 715   = wceq 1397   u. cun 3198   C_ wss 3200   ~Pcpw 3652

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  pwunim  4383