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Theorem disjdif 3495
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 3355 . 2  |-  ( A  i^i  B )  C_  A
2 inssdif0im 3490 . 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 1353    \ cdif 3126    i^i cin 3128    C_ wss 3129   (/)c0 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423
This theorem is referenced by:  ssdifin0  3504  difdifdirss  3507  fvsnun1  5709  fvsnun2  5710  phplem2  6847  unfiin  6919  xpfi  6923  sbthlem7  6956  sbthlemi8  6957  exmidfodomrlemim  7194  fihashssdif  10782  zfz1isolem1  10804  fsumlessfi  11452  fprodsplit1f  11626  setsfun  12480  setsfun0  12481  setsslid  12495
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