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Theorem eqimss 3221
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 3182 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
21simplbi 274 1 (𝐴 = 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wss 3141
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-in 3147  df-ss 3154
This theorem is referenced by:  eqimss2  3222  uneqin  3398  sssnr  3765  sssnm  3766  ssprr  3768  sstpr  3769  snsspw  3776  pwpwssunieq  3987  elpwuni  3988  disjeq2  3996  disjeq1  3999  pwne  4172  pwssunim  4296  poeq2  4312  seeq1  4351  seeq2  4352  trsucss  4435  onsucelsucr  4519  xp11m  5079  funeq  5248  fnresdm  5337  fssxp  5395  ffdm  5398  fcoi1  5408  fof  5450  dff1o2  5478  fvmptss2  5604  fvmptssdm  5613  fprg  5712  dff1o6  5790  tposeq  6262  el2oss1o  6458  nntri1  6511  nntri2or2  6513  nnsseleq  6516  infnninf  7136  infnninfOLD  7137  nninfwlpoimlemg  7187  exmidontri2or  7256  frec2uzf1od  10420  hashinfuni  10771  setsresg  12514  setsslid  12527  strle1g  12580  cncnpi  14024  hmeores  14111  limcimolemlt  14429  recnprss  14452  0nninf  15050  nninfall  15055
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