Theorem pwundifss 4287

Description: Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.)

Assertion

Ref	Expression
pwundifss	|- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ~P ( A u. B )

Proof of Theorem pwundifss

Step	Hyp	Ref	Expression
1		undif1ss 3499	. 2 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ( ~P ( A u. B ) u. ~P A )
2		pwunss 4285	. . . . 5 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3		unss 3311	. . . . 5 |- ( ( ~P A C_ ~P ( A u. B ) /\ ~P B C_ ~P ( A u. B ) ) <-> ( ~P A u. ~P B ) C_ ~P ( A u. B ) )
4	2, 3	mpbir 146	. . . 4 |- ( ~P A C_ ~P ( A u. B ) /\ ~P B C_ ~P ( A u. B ) )
5	4	simpli 111	. . 3 |- ~P A C_ ~P ( A u. B )
6		ssequn2 3310	. . 3 |- ( ~P A C_ ~P ( A u. B ) <-> ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B ) )
7	5, 6	mpbi 145	. 2 |- ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B )
8	1, 7	sseqtri 3191	1 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ~P ( A u. B )

Syntax hints:   /\ wa 104   = wceq 1353   \ cdif 3128   u. cun 3129   C_ wss 3131   ~Pcpw 3577

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579

This theorem is referenced by: (None)