Theorem iundifdifd 28166

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 4363	. . . . 5 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4350	. . . . . 6 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3580	. . . . 5 |- ( O \ |^| A ) = ( O \ |^|_ x e. A x )
4	1, 3	eqtr4i 2454	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^| A )
5	4	difeq2i 3580	. . 3 |- ( O \ U_ x e. A ( O \ x ) ) = ( O \ ( O \ |^| A ) )
6		intssuni2 4278	. . . . 5 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
7		unipw 4667	. . . . 5 |- U. ~P O = O
8	6, 7	syl6sseq 3510	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ O )
9		dfss4 3707	. . . 4 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
10	8, 9	sylib 199	. . 3 |- ( ( A C_ ~P O /\ A =/= (/) ) -> ( O \ ( O \ |^| A ) ) = |^| A )
11	5, 10	syl5req 2476	. 2 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )
12	11	ex 435	1 |- ( A C_ ~P O -> ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   =/= wne 2618   \ cdif 3433   C_ wss 3436   (/)c0 3761   ~Pcpw 3979   U.cuni 4216   |^|cint 4252   U_ciun 4296   |^|_ciin 4297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-pw 3981  df-sn 3997  df-pr 3999  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299

This theorem is referenced by:  sigaclci  28949