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Theorem difin0ss 3616
 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 3564 . 2
2 iman 413 . . . . . 6
3 elin 3219 . . . . . . . 8
4 eldif 3221 . . . . . . . . 9
54anbi1i 676 . . . . . . . 8
63, 5bitri 240 . . . . . . 7
7 ancom 437 . . . . . . 7
8 annim 414 . . . . . . . 8
98anbi2i 675 . . . . . . 7
106, 7, 93bitr2i 264 . . . . . 6
112, 10xchbinxr 302 . . . . 5
12 ax-2 6 . . . . 5
1311, 12sylbir 204 . . . 4
1413al2imi 1561 . . 3
15 dfss2 3262 . . 3
16 dfss2 3262 . . 3
1714, 15, 163imtr4g 261 . 2
181, 17sylbi 187 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 358  wal 1540   wceq 1642   wcel 1710   cdif 3206   cin 3208   wss 3257  c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by: (None)
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