Theorem iundifdif 28174

Description: The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 28173 (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 4364	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4351	. . . . 5 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3581	. . . 4 |- ( O \ |^| A ) = ( O \ |^|_ x e. A x )
4	1, 3	eqtr4i 2455	. . 3 |- U_ x e. A ( O \ x ) = ( O \ |^| A )
5	4	difeq2i 3581	. 2 |- ( O \ U_ x e. A ( O \ x ) ) = ( O \ ( O \ |^| A ) )
6		iundifdif.2	. . . . 5 |- A C_ ~P O
7	6	jctl 544	. . . 4 |- ( A =/= (/) -> ( A C_ ~P O /\ A =/= (/) ) )
8		intssuni2 4279	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
9		unipw 4669	. . . . . 6 |- U. ~P O = O
10	9	sseq2i 3490	. . . . 5 |- ( |^| A C_ U. ~P O <-> |^| A C_ O )
11	10	biimpi 198	. . . 4 |- ( |^| A C_ U. ~P O -> |^| A C_ O )
12	7, 8, 11	3syl 18	. . 3 |- ( A =/= (/) -> |^| A C_ O )
13		dfss4 3708	. . 3 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
14	12, 13	sylib 200	. 2 |- ( A =/= (/) -> ( O \ ( O \ |^| A ) ) = |^| A )
15	5, 14	syl5req 2477	1 |- ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1438   e. wcel 1869   =/= wne 2619   _Vcvv 3082   \ cdif 3434   C_ wss 3437   (/)c0 3762   ~Pcpw 3980   U.cuni 4217   |^|cint 4253   U_ciun 4297   |^|_ciin 4298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-pw 3982  df-sn 3998  df-pr 4000  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300

This theorem is referenced by: (None)