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Theorem unabs 3230
Description: Absorption law for union. (Contributed by NM, 16-Apr-2006.)
Assertion
Ref Expression
unabs (𝐴 ∪ (𝐴𝐵)) = 𝐴

Proof of Theorem unabs
StepHypRef Expression
1 inss1 3220 . 2 (𝐴𝐵) ⊆ 𝐴
2 ssequn2 3173 . 2 ((𝐴𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴𝐵)) = 𝐴)
31, 2mpbi 143 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cun 2997  cin 2998  wss 2999
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-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-un 3003  df-in 3005  df-ss 3012
This theorem is referenced by: (None)
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