MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snsspr1 Structured version   Unicode version

Theorem snsspr1 3939
Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
Assertion
Ref Expression
snsspr1  |-  { A }  C_  { A ,  B }

Proof of Theorem snsspr1
StepHypRef Expression
1 ssun1 3502 . 2  |-  { A }  C_  ( { A }  u.  { B } )
2 df-pr 3813 . 2  |-  { A ,  B }  =  ( { A }  u.  { B } )
31, 2sseqtr4i 3373 1  |-  { A }  C_  { A ,  B }
Colors of variables: wff set class
Syntax hints:    u. cun 3310    C_ wss 3312   {csn 3806   {cpr 3807
This theorem is referenced by:  snsstp1  3941  uniop  4451  op1stb  4750  rankopb  7770  ltrelxr  9131  2strbas  13558  algsca  13594  phlvsca  13604  prdssca  13671  prdshom  13681  imassca  13737  ipobas  14573  ipolerval  14574  lspprid1  16065  lsppratlem3  16213  lsppratlem4  16214  constr3pthlem1  21634  ex-dif  21723  ex-un  21724  ex-in  21725  coinflippv  24733  subfacp1lem2a  24858  altopthsn  25798  rankaltopb  25816  dvh3dim3N  32184  mapdindp2  32456  lspindp5  32505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-pr 3813
  Copyright terms: Public domain W3C validator