Theorem iundifdifd 26064

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 4346	. . . . 5 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4333	. . . . . 6 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3580	. . . . 5 |- ( O \ |^| A ) = ( O \ |^|_ x e. A x )
4	1, 3	eqtr4i 2486	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^| A )
5	4	difeq2i 3580	. . 3 |- ( O \ U_ x e. A ( O \ x ) ) = ( O \ ( O \ |^| A ) )
6		intssuni2 4262	. . . . 5 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
7		unipw 4651	. . . . 5 |- U. ~P O = O
8	6, 7	syl6sseq 3511	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ O )
9		dfss4 3693	. . . 4 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
10	8, 9	sylib 196	. . 3 |- ( ( A C_ ~P O /\ A =/= (/) ) -> ( O \ ( O \ |^| A ) ) = |^| A )
11	5, 10	syl5req 2508	. 2 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )
12	11	ex 434	1 |- ( A C_ ~P O -> ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   =/= wne 2648   \ cdif 3434   C_ wss 3437   (/)c0 3746   ~Pcpw 3969   U.cuni 4200   |^|cint 4237   U_ciun 4280   |^|_ciin 4281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-pw 3971  df-sn 3987  df-pr 3989  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283

This theorem is referenced by:  sigaclci  26721