Theorem iundifdif 26057

Description: The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 26056 (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 4338	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4325	. . . . 5 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3572	. . . 4 |- ( O \ |^| A ) = ( O \ |^|_ x e. A x )
4	1, 3	eqtr4i 2483	. . 3 |- U_ x e. A ( O \ x ) = ( O \ |^| A )
5	4	difeq2i 3572	. 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 4254	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
9		unipw 4643	. . . . . 6 |- U. ~P O = O
10	9	sseq2i 3482	. . . . 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 3685	. . 3 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
14	12, 13	sylib 196	. 2 |- ( A =/= (/) -> ( O \ ( O \ |^| A ) ) = |^| A )
15	5, 14	syl5req 2505	1 |- ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2644   _Vcvv 3071   \ cdif 3426   C_ wss 3429   (/)c0 3738   ~Pcpw 3961   U.cuni 4192   |^|cint 4229   U_ciun 4272   |^|_ciin 4273

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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632

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 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-pw 3963  df-sn 3979  df-pr 3981  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275

This theorem is referenced by: (None)