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Theorem inabs 4214
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 4131 . 2 𝐴 ⊆ (𝐴𝐵)
2 df-ss 3926 . 2 (𝐴 ⊆ (𝐴𝐵) ↔ (𝐴 ∩ (𝐴𝐵)) = 𝐴)
31, 2mpbi 229 1 (𝐴 ∩ (𝐴𝐵)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cun 3907  cin 3908  wss 3909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3446  df-un 3914  df-in 3916  df-ss 3926
This theorem is referenced by:  dfif5  4501  caragenuncllem  44648
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