Theorem pwssunim 4424

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 3402	. . . . 5 |- ( B C_ A <-> ( A u. B ) = A )
2		pweq 3688	. . . . . 6 |- ( ( A u. B ) = A -> ~P ( A u. B ) = ~P A )
3		eqimss 3302	. . . . . 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 3399	. . . . 5 |- ( A C_ B <-> ( A u. B ) = B )
7		pweq 3688	. . . . . 6 |- ( ( A u. B ) = B -> ~P ( A u. B ) = ~P B )
8		eqimss 3302	. . . . . 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 771	. . 3 |- ( ( B C_ A \/ A C_ B ) -> ( ~P ( A u. B ) C_ ~P A \/ ~P ( A u. B ) C_ ~P B ) )
12	11	orcoms 742	. 2 |- ( ( A C_ B \/ B C_ A ) -> ( ~P ( A u. B ) C_ ~P A \/ ~P ( A u. B ) C_ ~P B ) )
13		ssun 3408	. 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 720   = wceq 1402   u. cun 3218   C_ wss 3220   ~Pcpw 3685

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by:  pwunim  4426