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Theorem eqimss 3224
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 = 𝐵𝐴𝐵)

Proof of Theorem eqimss
StepHypRef Expression
1 eqss 3185 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
21simplbi 274 1 (𝐴 = 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wss 3144
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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157
This theorem is referenced by:  eqimss2  3225  uneqin  3401  sssnr  3768  sssnm  3769  ssprr  3771  sstpr  3772  snsspw  3779  pwpwssunieq  3990  elpwuni  3991  disjeq2  3999  disjeq1  4002  pwne  4175  pwssunim  4299  poeq2  4315  seeq1  4354  seeq2  4355  trsucss  4438  onsucelsucr  4522  xp11m  5082  funeq  5251  fnresdm  5340  fssxp  5398  ffdm  5401  fcoi1  5411  fof  5453  dff1o2  5481  fvmptss2  5607  fvmptssdm  5616  fprg  5715  dff1o6  5793  tposeq  6266  el2oss1o  6462  nntri1  6515  nntri2or2  6517  nnsseleq  6520  infnninf  7140  infnninfOLD  7141  nninfwlpoimlemg  7191  exmidontri2or  7260  frec2uzf1od  10424  hashinfuni  10775  setsresg  12518  setsslid  12531  strle1g  12584  cncnpi  14125  hmeores  14212  limcimolemlt  14530  recnprss  14553  0nninf  15151  nninfall  15156
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