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Theorem inssdif0 3617
 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.
StepHypRef Expression
1 elin 3219 . . . . . 6
21imbi1i 315 . . . . 5
3 iman 413 . . . . 5
42, 3bitri 240 . . . 4
5 eldif 3221 . . . . . 6
65anbi2i 675 . . . . 5
7 elin 3219 . . . . 5
8 anass 630 . . . . 5
96, 7, 83bitr4ri 269 . . . 4
104, 9xchbinx 301 . . 3
1110albii 1566 . 2
12 dfss2 3262 . 2
13 eq0 3564 . 2
1411, 12, 133bitr4i 268 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   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:  disjdif  3622
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