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Theorem eqimss 3078
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 3040 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
21simplbi 268 1  |-  ( A  =  B  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    C_ wss 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012
This theorem is referenced by:  eqimss2  3079  uneqin  3250  sssnr  3595  sssnm  3596  ssprr  3598  sstpr  3599  snsspw  3606  pwpwssunieq  3815  elpwuni  3816  disjeq2  3824  disjeq1  3827  pwne  3993  pwssunim  4109  poeq2  4125  seeq1  4164  seeq2  4165  trsucss  4248  onsucelsucr  4323  xp11m  4864  funeq  5029  fnresdm  5117  fssxp  5172  ffdm  5175  fcoi1  5185  fof  5227  dff1o2  5252  fvmptss2  5373  fvmptssdm  5381  fprg  5474  dff1o6  5547  tposeq  6004  nntri1  6249  nntri2or2  6251  nnsseleq  6254  infnninf  6795  frec2uzf1od  9801  hashinfuni  10173  el2oss1o  11770  0nninf  11776  nninfall  11783
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