ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqimss Unicode version

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  =  B  ->  A  C_  B )

Proof of Theorem eqimss
StepHypRef Expression
1 eqss 3185 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
21simplbi 274 1  |-  ( A  =  B  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    C_ 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  4178  pwssunim  4302  poeq2  4318  seeq1  4357  seeq2  4358  trsucss  4441  onsucelsucr  4525  xp11m  5085  funeq  5255  fnresdm  5344  fssxp  5402  ffdm  5405  fcoi1  5415  fof  5457  dff1o2  5485  fvmptss2  5612  fvmptssdm  5621  fprg  5720  dff1o6  5798  tposeq  6273  el2oss1o  6469  nntri1  6522  nntri2or2  6524  nnsseleq  6527  infnninf  7153  infnninfOLD  7154  nninfwlpoimlemg  7204  exmidontri2or  7273  frec2uzf1od  10439  hashinfuni  10792  setsresg  12553  setsslid  12566  strle1g  12621  cncnpi  14205  hmeores  14292  limcimolemlt  14610  recnprss  14633  0nninf  15232  nninfall  15237
  Copyright terms: Public domain W3C validator