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Theorem inindif 4075
 Description: The intersection of an intersection and a difference is empty. (Contributed by set.mm contributors, 10-Mar-2015.)
Assertion
Ref Expression
inindif ((AB) ∩ (A B)) =

Proof of Theorem inindif
StepHypRef Expression
1 df-dif 3215 . . 3 (A B) = (A ∩ ∼ B)
21ineq2i 3454 . 2 ((AB) ∩ (A B)) = ((AB) ∩ (A ∩ ∼ B))
3 inindi 3472 . 2 (A ∩ (B ∩ ∼ B)) = ((AB) ∩ (A ∩ ∼ B))
4 incompl 4073 . . . 4 (B ∩ ∼ B) =
54ineq2i 3454 . . 3 (A ∩ (B ∩ ∼ B)) = (A)
6 in0 3576 . . 3 (A) =
75, 6eqtri 2373 . 2 (A ∩ (B ∩ ∼ B)) =
82, 3, 73eqtr2i 2379 1 ((AB) ∩ (A B)) =
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∼ ccompl 3205   ∖ cdif 3206   ∩ cin 3208  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-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:  sbthlem1  6203
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