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Theorem difun2 3629
 Description: Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
difun2 ((AB) B) = (A B)

Proof of Theorem difun2
StepHypRef Expression
1 difundir 3508 . 2 ((AB) B) = ((A B) ∪ (B B))
2 difid 3618 . . 3 (B B) =
32uneq2i 3415 . 2 ((A B) ∪ (B B)) = ((A B) ∪ )
4 un0 3575 . 2 ((A B) ∪ ) = (A B)
51, 3, 43eqtri 2377 1 ((AB) B) = (A B)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∖ cdif 3206   ∪ cun 3207  ∅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-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  uneqdifeq  3638  difprsn1  3847  adj11  3889
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