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Theorem undifabs 3337
 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 3028 . . 3 𝐴𝐴
2 difss 3109 . . 3 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3158 . 2 (𝐴 ∪ (𝐴𝐵)) ⊆ 𝐴
4 ssun1 3146 . 2 𝐴 ⊆ (𝐴 ∪ (𝐴𝐵))
53, 4eqssi 3025 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1285   ∖ cdif 2980   ∪ cun 2981 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 2613  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996 This theorem is referenced by: (None)
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