Theorem inssdif0 3895

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

Proof of Theorem inssdif0

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3686	. . . . . 6
2	1	imbi1i 325	. . . . 5
3		iman 424	. . . . 5
4	2, 3	bitri 249	. . . 4
5		eldif 3485	. . . . . 6
6	5	anbi2i 694	. . . . 5
7		elin 3686	. . . . 5
8		anass 649	. . . . 5
9	6, 7, 8	3bitr4ri 278	. . . 4
10	4, 9	xchbinx 310	. . 3
11	10	albii 1640	. 2
12		dfss2 3492	. 2
13		eq0 3800	. 2
14	11, 12, 13	3bitr4i 277	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  \cdif 3472  i^icin 3474  C_wss 3475  c0 3784

This theorem is referenced by:  disjdif  3900  inf3lem3  8068  ssfin4  8711  isnrm2  19859  1stccnp  19963  llycmpkgen2  20051  ufileu  20420  fclscf  20526  flimfnfcls  20529  inindif  27414  opnbnd  30143  diophrw  30692  setindtr  30966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785