Theorem iundifdifd 28226

Description: The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Assertion

Ref	Expression
iundifdifd	|- ( A C_ ~P O -> ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) ) )

Distinct variable groups:   x, A   x, O

Proof of Theorem iundifdifd

Step	Hyp	Ref	Expression
1		iundif2 4359	. . . . 5 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4346	. . . . . 6 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3560	. . . . 5 |- ( O \ |^| A ) = ( O \ |^|_ x e. A x )
4	1, 3	eqtr4i 2487	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^| A )
5	4	difeq2i 3560	. . 3 |- ( O \ U_ x e. A ( O \ x ) ) = ( O \ ( O \ |^| A ) )
6		intssuni2 4274	. . . . 5 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
7		unipw 4664	. . . . 5 |- U. ~P O = O
8	6, 7	syl6sseq 3490	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ O )
9		dfss4 3689	. . . 4 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
10	8, 9	sylib 201	. . 3 |- ( ( A C_ ~P O /\ A =/= (/) ) -> ( O \ ( O \ |^| A ) ) = |^| A )
11	5, 10	syl5req 2509	. 2 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )
12	11	ex 440	1 |- ( A C_ ~P O -> ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1455   =/= wne 2633   \ cdif 3413   C_ wss 3416   (/)c0 3743   ~Pcpw 3963   U.cuni 4212   |^|cint 4248   U_ciun 4292   |^|_ciin 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-pw 3965  df-sn 3981  df-pr 3983  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295

This theorem is referenced by:  sigaclci  29003