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Theorem unabs 3167
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 3157 . 2 (𝐴𝐵) ⊆ 𝐴
2 ssequn2 3116 . 2 ((𝐴𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴𝐵)) = 𝐴)
31, 2mpbi 133 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1243  cun 2915  cin 2916  wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931
This theorem is referenced by: (None)
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