Theorem eqimss 3233

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

Step	Hyp	Ref	Expression
1		eqss 3194	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
2	1	simplbi 274	1 |- ( A = B -> A C_ B )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3153

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 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  eqimss2  3234  uneqin  3410  sssnr  3779  sssnm  3780  ssprr  3782  sstpr  3783  snsspw  3790  pwpwssunieq  4001  elpwuni  4002  disjeq2  4010  disjeq1  4013  pwne  4189  pwssunim  4315  poeq2  4331  seeq1  4370  seeq2  4371  trsucss  4454  onsucelsucr  4540  xp11m  5104  funeq  5274  fnresdm  5363  fssxp  5421  ffdm  5424  fcoi1  5434  fof  5476  dff1o2  5505  fvmptss2  5632  fvmptssdm  5642  fprg  5741  dff1o6  5819  tposeq  6300  el2oss1o  6496  nntri1  6549  nntri2or2  6551  nnsseleq  6554  infnninf  7183  infnninfOLD  7184  nninfwlpoimlemg  7234  exmidontri2or  7303  frec2uzf1od  10477  hashinfuni  10848  setsresg  12656  setsslid  12669  strle1g  12724  cncnpi  14396  hmeores  14483  limcimolemlt  14818  recnprss  14841  0nninf  15494  nninfall  15499