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Theorem undifabs 3627
 Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
Assertion
Ref Expression
undifabs (A ∪ (A B)) = A

Proof of Theorem undifabs
StepHypRef Expression
1 undif3 3515 . 2 (A ∪ (A B)) = ((AA) (B A))
2 unidm 3407 . . 3 (AA) = A
32difeq1i 3381 . 2 ((AA) (B A)) = (A (B A))
4 difdif 3392 . 2 (A (B A)) = A
51, 3, 43eqtri 2377 1 (A ∪ (A B)) = A
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∖ cdif 3206   ∪ cun 3207 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215 This theorem is referenced by:  dfif5  3674
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