Theorem difin0ss 3894

Description: Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Assertion

Ref	Expression
difin0ss

Proof of Theorem difin0ss

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eq0 3800	. 2
2		iman 424	. . . . . 6
3		elin 3686	. . . . . . . 8
4		eldif 3485	. . . . . . . . 9
5	4	anbi1i 695	. . . . . . . 8
6	3, 5	bitri 249	. . . . . . 7
7		ancom 450	. . . . . . 7
8		annim 425	. . . . . . . 8
9	8	anbi2i 694	. . . . . . 7
10	6, 7, 9	3bitr2i 273	. . . . . 6
11	2, 10	xchbinxr 311	. . . . 5
12		ax-2 7	. . . . 5
13	11, 12	sylbir 213	. . . 4
14	13	al2imi 1636	. . 3
15		dfss2 3492	. . 3
16		dfss2 3492	. . 3
17	14, 15, 16	3imtr4g 270	. 2
18	1, 17	sylbi 195	1

Syntax hints:  -.wn 3  ->wi 4  /\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:  tz7.7  4909  tfi  6688  lebnumlem3  21463

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