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Theorem difss 3197
Description: Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
Assertion
Ref Expression
difss  |-  ( A 
\  B )  C_  A

Proof of Theorem difss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldifi 3193 . 2  |-  ( x  e.  ( A  \  B )  ->  x  e.  A )
21ssriv 3096 1  |-  ( A 
\  B )  C_  A
Colors of variables: wff set class
Syntax hints:    \ cdif 3063    C_ wss 3066
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079
This theorem is referenced by:  difssd  3198  difss2  3199  ssdifss  3201  0dif  3429  undif1ss  3432  undifabs  3434  inundifss  3435  undifss  3438  unidif  3763  iunxdif2  3856  difexg  4064  reldif  4654  cnvdif  4940  resdif  5382  fndmdif  5518  swoer  6450  swoord1  6451  swoord2  6452  phplem2  6740  phpm  6752  unfiin  6807  sbthlem2  6839  sbthlemi4  6841  sbthlemi5  6842  difinfinf  6979  pinn  7110  niex  7113  dmaddpi  7126  dmmulpi  7127  lerelxr  7820  fisumss  11154  structcnvcnv  11964  strleund  12036  strleun  12037  strle1g  12038  discld  12294  exmid1stab  13184
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