Theorem inssdif0 3947

Description: Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Assertion

Ref	Expression
inssdif0	|- ( ( A i^i B ) C_ C <-> ( A i^i ( B \ C ) ) = (/) )

Proof of Theorem inssdif0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . . . . 6 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2	1	imbi1i 339	. . . . 5 |- ( ( x e. ( A i^i B ) -> x e. C ) <-> ( ( x e. A /\ x e. B ) -> x e. C ) )
3		iman 440	. . . . 5 |- ( ( ( x e. A /\ x e. B ) -> x e. C ) <-> -. ( ( x e. A /\ x e. B ) /\ -. x e. C ) )
4	2, 3	bitri 264	. . . 4 |- ( ( x e. ( A i^i B ) -> x e. C ) <-> -. ( ( x e. A /\ x e. B ) /\ -. x e. C ) )
5		eldif 3584	. . . . . 6 |- ( x e. ( B \ C ) <-> ( x e. B /\ -. x e. C ) )
6	5	anbi2i 730	. . . . 5 |- ( ( x e. A /\ x e. ( B \ C ) ) <-> ( x e. A /\ ( x e. B /\ -. x e. C ) ) )
7		elin 3796	. . . . 5 |- ( x e. ( A i^i ( B \ C ) ) <-> ( x e. A /\ x e. ( B \ C ) ) )
8		anass 681	. . . . 5 |- ( ( ( x e. A /\ x e. B ) /\ -. x e. C ) <-> ( x e. A /\ ( x e. B /\ -. x e. C ) ) )
9	6, 7, 8	3bitr4ri 293	. . . 4 |- ( ( ( x e. A /\ x e. B ) /\ -. x e. C ) <-> x e. ( A i^i ( B \ C ) ) )
10	4, 9	xchbinx 324	. . 3 |- ( ( x e. ( A i^i B ) -> x e. C ) <-> -. x e. ( A i^i ( B \ C ) ) )
11	10	albii 1747	. 2 |- ( A. x ( x e. ( A i^i B ) -> x e. C ) <-> A. x -. x e. ( A i^i ( B \ C ) ) )
12		dfss2 3591	. 2 |- ( ( A i^i B ) C_ C <-> A. x ( x e. ( A i^i B ) -> x e. C ) )
13		eq0 3929	. 2 |- ( ( A i^i ( B \ C ) ) = (/) <-> A. x -. x e. ( A i^i ( B \ C ) ) )
14	11, 12, 13	3bitr4i 292	1 |- ( ( A i^i B ) C_ C <-> ( A i^i ( B \ C ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  disjdif  4040  inf3lem3  8527  ssfin4  9132  isnrm2  21162  1stccnp  21265  llycmpkgen2  21353  ufileu  21723  fclscf  21829  flimfnfcls  21832  inindif  29353  opnbnd  32320  diophrw  37322  setindtr  37591