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Theorem difin0ss 3833
 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 3747 . 2
2 iman 426 . . . . . 6
3 elin 3617 . . . . . . . 8
4 eldif 3414 . . . . . . . . 9
54anbi1i 701 . . . . . . . 8
63, 5bitri 253 . . . . . . 7
7 ancom 452 . . . . . . 7
8 annim 427 . . . . . . . 8
98anbi2i 700 . . . . . . 7
106, 7, 93bitr2i 277 . . . . . 6
112, 10xchbinxr 313 . . . . 5
12 ax-2 7 . . . . 5
1311, 12sylbir 217 . . . 4
1413al2imi 1687 . . 3
15 dfss2 3421 . . 3
16 dfss2 3421 . . 3
1714, 15, 163imtr4g 274 . 2
181, 17sylbi 199 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 371  wal 1442   wceq 1444   wcel 1887   cdif 3401   cin 3403   wss 3404  c0 3731 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732 This theorem is referenced by:  tz7.7  5449  tfi  6680  lebnumlem3  21991  lebnumlem3OLD  21994
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