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Theorem undif2 3626
 Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3622). Part of proof of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
undif2 (A ∪ (B A)) = (AB)

Proof of Theorem undif2
StepHypRef Expression
1 uncom 3408 . 2 (A ∪ (B A)) = ((B A) ∪ A)
2 undif1 3625 . 2 ((B A) ∪ A) = (BA)
3 uncom 3408 . 2 (BA) = (AB)
41, 2, 33eqtri 2377 1 (A ∪ (B A)) = (AB)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∖ cdif 3206   ∪ cun 3207 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:  undif  3630  dfif5  3674  dflec2  6210
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