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Theorem difun2 3338
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 3233 . 2 ((𝐴𝐵) ∖ 𝐵) = ((𝐴𝐵) ∪ (𝐵𝐵))
2 difid 3328 . . 3 (𝐵𝐵) = ∅
32uneq2i 3133 . 2 ((𝐴𝐵) ∪ (𝐵𝐵)) = ((𝐴𝐵) ∪ ∅)
4 un0 3294 . 2 ((𝐴𝐵) ∪ ∅) = (𝐴𝐵)
51, 3, 43eqtri 2107 1 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1285  cdif 2979  cun 2980  c0 3267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268
This theorem is referenced by:  uneqdifeqim  3344  difprsn1  3544  orddif  4318  fisseneq  6474  dfn2  8420
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