Theorem iundifdifd 29380

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 4587	. . . . 5 |- U_ x e. A ( O \ x ) = ( O \ |^|_ x e. A x )
2		intiin 4574	. . . . . 6 |- |^| A = |^|_ x e. A x
3	2	difeq2i 3725	. . . . 5 |- ( O \ |^| A ) = ( O \ |^|_ x e. A x )
4	1, 3	eqtr4i 2647	. . . 4 |- U_ x e. A ( O \ x ) = ( O \ |^| A )
5	4	difeq2i 3725	. . 3 |- ( O \ U_ x e. A ( O \ x ) ) = ( O \ ( O \ |^| A ) )
6		intssuni2 4502	. . . . 5 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ U. ~P O )
7		unipw 4918	. . . . 5 |- U. ~P O = O
8	6, 7	syl6sseq 3651	. . . 4 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A C_ O )
9		dfss4 3858	. . . 4 |- ( |^| A C_ O <-> ( O \ ( O \ |^| A ) ) = |^| A )
10	8, 9	sylib 208	. . 3 |- ( ( A C_ ~P O /\ A =/= (/) ) -> ( O \ ( O \ |^| A ) ) = |^| A )
11	5, 10	syl5req 2669	. 2 |- ( ( A C_ ~P O /\ A =/= (/) ) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) )
12	11	ex 450	1 |- ( A C_ ~P O -> ( A =/= (/) -> |^| A = ( O \ U_ x e. A ( O \ x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   U_ciun 4520   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523

This theorem is referenced by:  sigaclci  30195