Theorem eqimss 3052

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 3015	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
2	1	simplbi 268	1 |- ( A = B -> A C_ B )

Syntax hints:   -> wi 4   = wceq 1285   C_ wss 2974

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987

This theorem is referenced by:  eqimss2  3053  uneqin  3222  sssnr  3553  sssnm  3554  ssprr  3556  sstpr  3557  snsspw  3564  elpwuni  3770  disjeq2  3778  disjeq1  3781  pwne  3942  pwssunim  4047  poeq2  4063  seeq1  4102  seeq2  4103  trsucss  4186  onsucelsucr  4260  xp11m  4789  funeq  4951  fnresdm  5039  fssxp  5089  ffdm  5092  fcoi1  5101  fof  5137  dff1o2  5162  fvmptss2  5279  fvmptssdm  5287  fprg  5378  dff1o6  5447  tposeq  5896  nntri1  6140  nntri2or2  6142  nnsseleq  6145  frec2uzf1od  9488  sizeinfuni  9801