Theorem eqimssi 4040

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 4002	. 2 ⊢ 𝐴 ⊆ 𝐴
2		eqimssi.1	. 2 ⊢ 𝐴 = 𝐵
3	1, 2	sseqtri 4016	1 ⊢ 𝐴 ⊆ 𝐵

Syntax hints:   = wceq 1534   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964

This theorem is referenced by:  funi  6585  fpr  7163  tz7.48-2  8463  trcl  9752  zorn2lem4  10523  zmin  12959  elfzo1  13715  om2uzf1oi  13951  0trrel  14961  sumsplit  15747  isumless  15824  rnglidl1  21128  frlmip  21712  ust0  24137  rrxprds  25330  rrxip  25331  ovoliunnul  25449  vitalilem5  25554  logtayl  26607  nbgr2vtx1edg  29176  nbuhgr2vtx1edgb  29178  mayetes3i  31552  cycpmconjslem2  32889  eulerpartlemsv2  33978  eulerpartlemsv3  33981  eulerpartlemv  33984  eulerpartlemb  33988  poimirlem9  37102  dvasin  37177  dmcoss3  37925  disjALTVid  38227  sticksstones17  41635  sticksstones18  41636  nna4b4nsq  42084  cnvrcl0  43055  corclrcl  43137  trclrelexplem  43141  cotrcltrcl  43155  he0  43214  dvsid  43768  binomcxplemnotnn0  43793  fourierdlem62  45556  fourierdlem66  45560