 Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inin Structured version   Visualization version   GIF version

Theorem inin 29659
 Description: Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
inin (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem inin
StepHypRef Expression
1 in13 3969 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐵 ∩ (𝐴𝐴))
2 inidm 3965 . . 3 (𝐴𝐴) = 𝐴
32ineq2i 3954 . 2 (𝐵 ∩ (𝐴𝐴)) = (𝐵𝐴)
4 incom 3948 . 2 (𝐵𝐴) = (𝐴𝐵)
51, 3, 43eqtri 2786 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∩ cin 3714 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722 This theorem is referenced by:  measinb2  30595
 Copyright terms: Public domain W3C validator