Theorem iundifdif 28226

Description: The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 28225. (Contributed by Thierry Arnoux, 4-Sep-2016.)

Hypotheses

Ref	Expression
iundifdif.o	|- O e. _V
iundifdif.2	|- A C_ ~P O

Assertion

Ref	Expression
iundifdif	|- ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )

Distinct variable groups:   x, A   x, O

Proof of Theorem iundifdif

Step	Hyp	Ref	Expression
1		iundif2 4358	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4345	. . . . 5 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3559	. . . 4 |- ( O \ |^| A ) = ( O \ |^|_ x e. A x )
4	1, 3	eqtr4i 2486	. . 3 |- U_ x e. A ( O \ x ) = ( O \ |^| A )
5	4	difeq2i 3559	. 2 |- ( O \ U_ x e. A ( O \ x ) ) = ( O \ ( O \ |^| A ) )
6		iundifdif.2	. . . . 5 |- A C_ ~P O
7	6	jctl 548	. . . 4 |- ( A =/= (/) -> ( A C_ ~P O /\ A =/= (/) ) )
8		intssuni2 4273	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
9		unipw 4663	. . . . . 6 |- U. ~P O = O
10	9	sseq2i 3468	. . . . 5 |- ( |^| A C_ U. ~P O <-> |^| A C_ O )
11	10	biimpi 199	. . . 4 |- ( |^| A C_ U. ~P O -> |^| A C_ O )
12	7, 8, 11	3syl 18	. . 3 |- ( A =/= (/) -> |^| A C_ O )
13		dfss4 3688	. . 3 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
14	12, 13	sylib 201	. 2 |- ( A =/= (/) -> ( O \ ( O \ |^| A ) ) = |^| A )
15	5, 14	syl5req 2508	1 |- ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )

Syntax hints:   -> wi 4   /\ wa 375   = wceq 1454   e. wcel 1897   =/= wne 2632   _Vcvv 3056   \ cdif 3412   C_ wss 3415   (/)c0 3742   ~Pcpw 3962   U.cuni 4211   |^|cint 4247   U_ciun 4291   |^|_ciin 4292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-pw 3964  df-sn 3980  df-pr 3982  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294

This theorem is referenced by: (None)