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Theorem inabs 3193
Description: Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
Assertion
Ref Expression
inabs (𝐴 ∩ (𝐴𝐵)) = 𝐴

Proof of Theorem inabs
StepHypRef Expression
1 ssun1 3131 . 2 𝐴 ⊆ (𝐴𝐵)
2 df-ss 2956 . 2 (𝐴 ⊆ (𝐴𝐵) ↔ (𝐴 ∩ (𝐴𝐵)) = 𝐴)
31, 2mpbi 137 1 (𝐴 ∩ (𝐴𝐵)) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1257  cun 2940  cin 2941  wss 2942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-in 2949  df-ss 2956
This theorem is referenced by: (None)
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