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Theorem snsstp2 4030
Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
Assertion
Ref Expression
snsstp2  |-  { B }  C_  { A ,  B ,  C }

Proof of Theorem snsstp2
StepHypRef Expression
1 snsspr2 4028 . . 3  |-  { B }  C_  { A ,  B }
2 ssun1 3524 . . 3  |-  { A ,  B }  C_  ( { A ,  B }  u.  { C } )
31, 2sstri 3370 . 2  |-  { B }  C_  ( { A ,  B }  u.  { C } )
4 df-tp 3887 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
53, 4sseqtr4i 3394 1  |-  { B }  C_  { A ,  B ,  C }
Colors of variables: wff setvar class
Syntax hints:    u. cun 3331    C_ wss 3333   {csn 3882   {cpr 3884   {ctp 3886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-v 2979  df-un 3338  df-in 3340  df-ss 3347  df-pr 3885  df-tp 3887
This theorem is referenced by:  fr3nr  6396  rngplusg  14292  srngplusg  14300  lmodplusg  14309  ipsaddg  14316  ipsvsca  14319  phlplusg  14326  topgrpplusg  14334  otpstset  14341  odrngplusg  14352  odrngle  14355  prdsplusg  14401  prdsvsca  14403  prdsle  14405  imasplusg  14460  imasvsca  14463  imasle  14466  fuchom  14876  setchomfval  14952  catchomfval  14971  xpchomfval  14994  psrplusg  17457  psrvscafval  17466  cnfldadd  17828  cnfldle  17832  trkgdist  22912  algaddg  29541
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