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Theorem difss 3172
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 3168 . 2  |-  ( x  e.  ( A  \  B )  ->  x  e.  A )
21ssriv 3071 1  |-  ( A 
\  B )  C_  A
Colors of variables: wff set class
Syntax hints:    \ cdif 3038    C_ wss 3041
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
This theorem is referenced by:  difssd  3173  difss2  3174  ssdifss  3176  0dif  3404  undif1ss  3407  undifabs  3409  inundifss  3410  undifss  3413  unidif  3738  iunxdif2  3831  difexg  4039  reldif  4629  cnvdif  4915  resdif  5357  fndmdif  5493  swoer  6425  swoord1  6426  swoord2  6427  phplem2  6715  phpm  6727  unfiin  6782  sbthlem2  6814  sbthlemi4  6816  sbthlemi5  6817  difinfinf  6954  pinn  7085  niex  7088  dmaddpi  7101  dmmulpi  7102  lerelxr  7795  fisumss  11129  structcnvcnv  11902  strleund  11974  strleun  11975  strle1g  11976  discld  12232  exmid1stab  13122
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