Theorem sseqtrri 3137

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)

Hypotheses

Ref	Expression
sseqtrri.1	|- A C_ B
sseqtrri.2	|- C = B

Assertion

Ref	Expression
sseqtrri	|- A C_ C

Proof of Theorem sseqtrri

Step	Hyp	Ref	Expression
1		sseqtrri.1	. 2 |- A C_ B
2		sseqtrri.2	. . 3 |- C = B
3	2	eqcomi 2144	. 2 |- B = C
4	1, 3	sseqtri 3136	1 |- A C_ C

Syntax hints:   = wceq 1332   C_ wss 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089

This theorem is referenced by:  eqimss2i  3159  difdif2ss  3338  snsspr1  3676  snsspr2  3677  snsstp1  3678  snsstp2  3679  snsstp3  3680  prsstp12  3681  prsstp13  3682  prsstp23  3683  iunxdif2  3869  pwpwssunieq  3909  sssucid  4345  opabssxp  4621  dmresi  4882  cnvimass  4910  ssrnres  4989  cnvcnv  4999  cnvssrndm  5068  dmmpossx  6105  tfrcllemssrecs  6257  sucinc  6349  mapex  6556  exmidpw  6810  casefun  6978  djufun  6997  ressxr  7833  ltrelxr  7849  nnssnn0  9004  un0addcl  9034  un0mulcl  9035  nn0ssxnn0  9067  fzssnn  9879  fzossnn0  9983  isumclim3  11224  isprm3  11835  phimullem  11937  tgvalex  12258  cnrest2  12444  qtopbasss  12729  tgqioo  12755