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Theorem undifabs 3359
Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
Assertion
Ref Expression
undifabs (𝐴 ∪ (𝐴𝐵)) = 𝐴

Proof of Theorem undifabs
StepHypRef Expression
1 ssid 3044 . . 3 𝐴𝐴
2 difss 3126 . . 3 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3175 . 2 (𝐴 ∪ (𝐴𝐵)) ⊆ 𝐴
4 ssun1 3163 . 2 𝐴 ⊆ (𝐴 ∪ (𝐴𝐵))
53, 4eqssi 3041 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cdif 2996  cun 2997
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012
This theorem is referenced by: (None)
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