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Theorem inabs 4272
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 4188 . 2 𝐴 ⊆ (𝐴𝐵)
2 dfss2 3981 . 2 (𝐴 ⊆ (𝐴𝐵) ↔ (𝐴 ∩ (𝐴𝐵)) = 𝐴)
31, 2mpbi 230 1 (𝐴 ∩ (𝐴𝐵)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cun 3961  cin 3962  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-in 3970  df-ss 3980
This theorem is referenced by:  dfif5  4547  caragenuncllem  46468
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