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Theorem disjdif 3405
Description: A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
disjdif  |-  ( A  i^i  ( B  \  A ) )  =  (/)

Proof of Theorem disjdif
StepHypRef Expression
1 inss1 3266 . 2  |-  ( A  i^i  B )  C_  A
2 inssdif0im 3400 . 2  |-  ( ( A  i^i  B ) 
C_  A  ->  ( A  i^i  ( B  \  A ) )  =  (/) )
31, 2ax-mp 5 1  |-  ( A  i^i  ( B  \  A ) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1316    \ cdif 3038    i^i cin 3040    C_ wss 3041   (/)c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by:  ssdifin0  3414  difdifdirss  3417  fvsnun1  5585  fvsnun2  5586  phplem2  6715  unfiin  6782  xpfi  6786  sbthlem7  6819  sbthlemi8  6820  exmidfodomrlemim  7025  fihashssdif  10532  zfz1isolem1  10551  fsumlessfi  11197  setsfun  11921  setsfun0  11922  setsslid  11936
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