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Theorem eqimss 3121
Description: Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
eqimss  |-  ( A  =  B  ->  A  C_  B )

Proof of Theorem eqimss
StepHypRef Expression
1 eqss 3082 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
21simplbi 272 1  |-  ( A  =  B  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    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-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  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-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054
This theorem is referenced by:  eqimss2  3122  uneqin  3297  sssnr  3650  sssnm  3651  ssprr  3653  sstpr  3654  snsspw  3661  pwpwssunieq  3871  elpwuni  3872  disjeq2  3880  disjeq1  3883  pwne  4054  pwssunim  4176  poeq2  4192  seeq1  4231  seeq2  4232  trsucss  4315  onsucelsucr  4394  xp11m  4947  funeq  5113  fnresdm  5202  fssxp  5260  ffdm  5263  fcoi1  5273  fof  5315  dff1o2  5340  fvmptss2  5464  fvmptssdm  5473  fprg  5571  dff1o6  5645  tposeq  6112  nntri1  6360  nntri2or2  6362  nnsseleq  6365  infnninf  6990  frec2uzf1od  10147  hashinfuni  10491  setsresg  11924  setsslid  11936  strle1g  11976  cncnpi  12324  hmeores  12411  limcimolemlt  12729  recnprss  12752  el2oss1o  13115  0nninf  13124  nninfall  13131
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