Theorem eqimssi 3946

Description: Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)

Hypothesis

Ref	Expression
eqimssi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
eqimssi	⊢ 𝐴 ⊆ 𝐵

Proof of Theorem eqimssi

Step	Hyp	Ref	Expression
1		ssid 3910	. 2 ⊢ 𝐴 ⊆ 𝐴
2		eqimssi.1	. 2 ⊢ 𝐴 = 𝐵
3	1, 2	sseqtri 3924	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1522   ⊆ wss 3859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-in 3866  df-ss 3874

This theorem is referenced by:  funi  6257  fpr  6779  tz7.48-2  7929  trcl  9016  zorn2lem4  9767  zmin  12193  elfzo1  12937  om2uzf1oi  13171  0trrel  14175  sumsplit  14956  isumless  15033  frlmip  20604  ust0  22511  rrxprds  23675  rrxip  23676  ovoliunnul  23791  vitalilem5  23896  logtayl  24924  nbgr2vtx1edg  26815  nbuhgr2vtx1edgb  26817  mayetes3i  29197  cycpmconjslem2  30435  eulerpartlemsv2  31233  eulerpartlemsv3  31236  eulerpartlemv  31239  eulerpartlemb  31243  poimirlem9  34432  dvasin  34509  dmcoss3  35224  disjALTVid  35516  cnvrcl0  39470  corclrcl  39537  trclrelexplem  39541  cotrcltrcl  39555  he0  39615  dvsid  40201  binomcxplemnotnn0  40226  fourierdlem62  41995  fourierdlem66  41999