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Theorem undifabs 3514
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 3190 . . 3 𝐴𝐴
2 difss 3276 . . 3 (𝐴𝐵) ⊆ 𝐴
31, 2unssi 3325 . 2 (𝐴 ∪ (𝐴𝐵)) ⊆ 𝐴
4 ssun1 3313 . 2 𝐴 ⊆ (𝐴 ∪ (𝐴𝐵))
53, 4eqssi 3186 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1364  cdif 3141  cun 3142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157
This theorem is referenced by:  exmid1stab  4226
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