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

Theorem difeqri 3431
Description: Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypothesis
Ref Expression
difeqri.1  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  C
)
Assertion
Ref Expression
difeqri  |-  ( A 
\  B )  =  C
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem difeqri
StepHypRef Expression
1 eldif 3294 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 difeqri.1 . . 3  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  <->  x  e.  C
)
31, 2bitri 241 . 2  |-  ( x  e.  ( A  \  B )  <->  x  e.  C )
43eqriv 2405 1  |-  ( A 
\  B )  =  C
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    \ cdif 3281
This theorem is referenced by:  difdif  3437  ddif  3443  dfss4  3539  difin  3542  difab  3574
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-dif 3287
  Copyright terms: Public domain W3C validator