Theorem pwssunim 4214

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 3254	. . . . 5 |- ( B C_ A <-> ( A u. B ) = A )
2		pweq 3518	. . . . . 6 |- ( ( A u. B ) = A -> ~P ( A u. B ) = ~P A )
3		eqimss 3156	. . . . . 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 120	. . . 4 |- ( B C_ A -> ~P ( A u. B ) C_ ~P A )
6		ssequn1 3251	. . . . 5 |- ( A C_ B <-> ( A u. B ) = B )
7		pweq 3518	. . . . . 6 |- ( ( A u. B ) = B -> ~P ( A u. B ) = ~P B )
8		eqimss 3156	. . . . . 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 120	. . . 4 |- ( A C_ B -> ~P ( A u. B ) C_ ~P B )
11	5, 10	orim12i 749	. . 3 |- ( ( B C_ A \/ A C_ B ) -> ( ~P ( A u. B ) C_ ~P A \/ ~P ( A u. B ) C_ ~P B ) )
12	11	orcoms 720	. 2 |- ( ( A C_ B \/ B C_ A ) -> ( ~P ( A u. B ) C_ ~P A \/ ~P ( A u. B ) C_ ~P B ) )
13		ssun 3260	. 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 698   = wceq 1332   u. cun 3074   C_ wss 3076   ~Pcpw 3515

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517

This theorem is referenced by:  pwunim  4216