Theorem eqimssi 3800

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 3765	. 2 ⊢ 𝐴 ⊆ 𝐴
2		eqimssi.1	. 2 ⊢ 𝐴 = 𝐵
3	1, 2	sseqtri 3778	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1632   ⊆ wss 3715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729

This theorem is referenced by:  funi  6081  fpr  6584  tz7.48-2  7706  trcl  8777  zorn2lem4  9513  zmin  11977  elfzo1  12712  om2uzf1oi  12946  0trrel  13921  sumsplit  14698  isumless  14776  frlmip  20319  ust0  22224  rrxprds  23377  rrxip  23378  ovoliunnul  23475  vitalilem5  23580  logtayl  24605  nbgr2vtx1edg  26445  nbuhgr2vtx1edgb  26447  mayetes3i  28897  eulerpartlemsv2  30729  eulerpartlemsv3  30732  eulerpartlemv  30735  eulerpartlemb  30739  poimirlem9  33731  dvasin  33809  dmcoss3  34526  cnvrcl0  38434  corclrcl  38501  trclrelexplem  38505  cotrcltrcl  38519  he0  38580  idhe  38583  dvsid  39032  binomcxplemnotnn0  39057  fourierdlem62  40888  fourierdlem66  40892