MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfin4 Unicode version

Theorem dfin4 3517
Description: Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)
Assertion
Ref Expression
dfin4  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )

Proof of Theorem dfin4
StepHypRef Expression
1 inss1 3497 . . 3  |-  ( A  i^i  B )  C_  A
2 dfss4 3511 . . 3  |-  ( ( A  i^i  B ) 
C_  A  <->  ( A  \  ( A  \  ( A  i^i  B ) ) )  =  ( A  i^i  B ) )
31, 2mpbi 200 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  i^i  B
)
4 difin 3514 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
54difeq2i 3398 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  \  ( A  \  B ) )
63, 5eqtr3i 2402 1  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    \ cdif 3253    i^i cin 3255    C_ wss 3256
This theorem is referenced by:  indif  3519  cnvin  5212  imain  5462  resin  5630  elcls  17053  cmmbl  19289  mbfeqalem  19394  itg1addlem4  19451  itg1addlem5  19452  inelsiga  24307  stoweidlem50  27460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rab 2651  df-v 2894  df-dif 3259  df-in 3263  df-ss 3270
  Copyright terms: Public domain W3C validator