MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inabs Structured version   Visualization version   GIF version

Theorem inabs 4207
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 4124 . 2 𝐴 ⊆ (𝐴𝐵)
2 df-ss 3927 . 2 (𝐴 ⊆ (𝐴𝐵) ↔ (𝐴 ∩ (𝐴𝐵)) = 𝐴)
31, 2mpbi 233 1 (𝐴 ∩ (𝐴𝐵)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cun 3908  cin 3909  wss 3910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-un 3915  df-in 3917  df-ss 3927
This theorem is referenced by:  dfif5  4456  caragenuncllem  42944
  Copyright terms: Public domain W3C validator