Theorem iundifdif 27295

Description: The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 27294 (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 4378	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4365	. . . . 5 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3601	. . . 4 |- ( O \ |^| A ) = ( O \ |^|_ x e. A x )
4	1, 3	eqtr4i 2473	. . 3 |- U_ x e. A ( O \ x ) = ( O \ |^| A )
5	4	difeq2i 3601	. 2 |- ( O \ U_ x e. A ( O \ x ) ) = ( O \ ( O \ |^| A ) )
6		iundifdif.2	. . . . 5 |- A C_ ~P O
7	6	jctl 541	. . . 4 |- ( A =/= (/) -> ( A C_ ~P O /\ A =/= (/) ) )
8		intssuni2 4293	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
9		unipw 4683	. . . . . 6 |- U. ~P O = O
10	9	sseq2i 3511	. . . . 5 |- ( |^| A C_ U. ~P O <-> |^| A C_ O )
11	10	biimpi 194	. . . 4 |- ( |^| A C_ U. ~P O -> |^| A C_ O )
12	7, 8, 11	3syl 20	. . 3 |- ( A =/= (/) -> |^| A C_ O )
13		dfss4 3714	. . 3 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
14	12, 13	sylib 196	. 2 |- ( A =/= (/) -> ( O \ ( O \ |^| A ) ) = |^| A )
15	5, 14	syl5req 2495	1 |- ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1381   e. wcel 1802   =/= wne 2636   _Vcvv 3093   \ cdif 3455   C_ wss 3458   (/)c0 3767   ~Pcpw 3993   U.cuni 4230   |^|cint 4267   U_ciun 4311   |^|_ciin 4312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-pw 3995  df-sn 4011  df-pr 4013  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314

This theorem is referenced by: (None)