Theorem iundifdif 27640

Description: The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 27639 (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 4382	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4369	. . . . 5 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3605	. . . 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 3605	. 2 |- ( O \ U_ x e. A ( O \ x ) ) = ( O \ ( O \ |^| A ) )
6		iundifdif.2	. . . . 5 |- A C_ ~P O
7	6	jctl 539	. . . 4 |- ( A =/= (/) -> ( A C_ ~P O /\ A =/= (/) ) )
8		intssuni2 4297	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
9		unipw 4687	. . . . . 6 |- U. ~P O = O
10	9	sseq2i 3514	. . . . 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 3729	. . 3 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
14	12, 13	sylib 196	. 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 367   = wceq 1398   e. wcel 1823   =/= wne 2649   _Vcvv 3106   \ cdif 3458   C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235   |^|cint 4271   U_ciun 4315   |^|_ciin 4316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001  df-sn 4017  df-pr 4019  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318

This theorem is referenced by: (None)