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Theorem inabs 4266
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 4178 . 2 𝐴 ⊆ (𝐴𝐵)
2 dfss2 3969 . 2 (𝐴 ⊆ (𝐴𝐵) ↔ (𝐴 ∩ (𝐴𝐵)) = 𝐴)
31, 2mpbi 230 1 (𝐴 ∩ (𝐴𝐵)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cun 3949  cin 3950  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-in 3958  df-ss 3968
This theorem is referenced by:  dfif5  4542  caragenuncllem  46527
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