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

Theorem indif 3747
Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
indif  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)

Proof of Theorem indif
StepHypRef Expression
1 dfin4 3745 . 2  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  ( A  \  ( A  \  B ) ) )
2 dfin4 3745 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
32difeq2i 3615 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  ( A 
\  ( A  \  B ) ) )
4 difin 3742 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
51, 3, 43eqtr2i 2492 1  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395    \ cdif 3468    i^i cin 3470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485
This theorem is referenced by:  resdif  5842  kmlem11  8557  psgndiflemB  18763
  Copyright terms: Public domain W3C validator