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

Theorem indif 4268
Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
indif (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem indif
StepHypRef Expression
1 dfin4 4266 . 2 (𝐴 ∩ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
2 dfin4 4266 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
32difeq2i 4115 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴𝐵)))
4 difin 4260 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
51, 3, 43eqtr2i 2759 1 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cdif 3941  cin 3943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-in 3951  df-ss 3961
This theorem is referenced by:  resdif  6859  kmlem11  10185  psgndiflemB  21549
  Copyright terms: Public domain W3C validator