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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.
StepHypRef Expression
1 eq0 3800 . 2
2 iman 424 . . . . . 6
3 elin 3686 . . . . . . . 8
4 eldif 3485 . . . . . . . . 9
54anbi1i 695 . . . . . . . 8
63, 5bitri 249 . . . . . . 7
7 ancom 450 . . . . . . 7
8 annim 425 . . . . . . . 8
98anbi2i 694 . . . . . . 7
106, 7, 93bitr2i 273 . . . . . 6
112, 10xchbinxr 311 . . . . 5
12 ax-2 7 . . . . 5
1311, 12sylbir 213 . . . 4
1413al2imi 1636 . . 3
15 dfss2 3492 . . 3
16 dfss2 3492 . . 3
1714, 15, 163imtr4g 270 . 2
181, 17sylbi 195 1
 Colors of variables: wff setvar class 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
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