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Theorem difun2 3446
Description: Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
difun2 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Proof of Theorem difun2
StepHypRef Expression
1 difundir 3333 . 2 ((𝐴𝐵) ∖ 𝐵) = ((𝐴𝐵) ∪ (𝐵𝐵))
2 difid 3435 . . 3 (𝐵𝐵) = ∅
32uneq2i 3231 . 2 ((𝐴𝐵) ∪ (𝐵𝐵)) = ((𝐴𝐵) ∪ ∅)
4 un0 3400 . 2 ((𝐴𝐵) ∪ ∅) = (𝐴𝐵)
51, 3, 43eqtri 2165 1 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1332  cdif 3072  cun 3073  c0 3367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368
This theorem is referenced by:  uneqdifeqim  3452  difprsn1  3666  orddif  4469  fisseneq  6827  dfn2  9013
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