Theorem iundifdifd 27088

Description: The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Assertion

Ref	Expression
iundifdifd	|- ( A C_ ~P O -> ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) ) )

Distinct variable groups:   x, A   x, O

Proof of Theorem iundifdifd

Step	Hyp	Ref	Expression
1		iundif2 4385	. . . . 5 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4372	. . . . . 6 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3612	. . . . 5 |- ( O \ |^| A ) = ( O \ |^|_ x e. A x )
4	1, 3	eqtr4i 2492	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^| A )
5	4	difeq2i 3612	. . 3 |- ( O \ U_ x e. A ( O \ x ) ) = ( O \ ( O \ |^| A ) )
6		intssuni2 4300	. . . . 5 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
7		unipw 4690	. . . . 5 |- U. ~P O = O
8	6, 7	syl6sseq 3543	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ O )
9		dfss4 3725	. . . 4 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
10	8, 9	sylib 196	. . 3 |- ( ( A C_ ~P O /\ A =/= (/) ) -> ( O \ ( O \ |^| A ) ) = |^| A )
11	5, 10	syl5req 2514	. 2 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )
12	11	ex 434	1 |- ( A C_ ~P O -> ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   =/= wne 2655   \ cdif 3466   C_ wss 3469   (/)c0 3778   ~Pcpw 4003   U.cuni 4238   |^|cint 4275   U_ciun 4318   |^|_ciin 4319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-pw 4005  df-sn 4021  df-pr 4023  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321

This theorem is referenced by:  sigaclci  27758