MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snsspr2 Structured version   Unicode version

Theorem snsspr2 4153
Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
Assertion
Ref Expression
snsspr2  |-  { B }  C_  { A ,  B }

Proof of Theorem snsspr2
StepHypRef Expression
1 ssun2 3636 . 2  |-  { B }  C_  ( { A }  u.  { B } )
2 df-pr 4005 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
31, 2sseqtr4i 3503 1  |-  { B }  C_  { A ,  B }
Colors of variables: wff setvar class
Syntax hints:    u. cun 3440    C_ wss 3442   {csn 4002   {cpr 4004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-un 3447  df-in 3449  df-ss 3456  df-pr 4005
This theorem is referenced by:  snsstp2  4155  ord3ex  4615  ltrelxr  9694  2strop  15185  phlip  15233  prdsco  15316  ipotset  16345  lsppratlem4  18299  constr3pthlem1  25219  ex-res  25727  subfacp1lem2a  29682  dvh3dim3N  34716  algvsca  35737  corclrcl  35928  gsumpr  38892
  Copyright terms: Public domain W3C validator