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Theorem snsstp2 3767
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 3765 . . 3  |-  { B }  C_  { A ,  B }
2 ssun1 3338 . . 3  |-  { A ,  B }  C_  ( { A ,  B }  u.  { C } )
31, 2sstri 3188 . 2  |-  { B }  C_  ( { A ,  B }  u.  { C } )
4 df-tp 3648 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
53, 4sseqtr4i 3211 1  |-  { B }  C_  { A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    u. cun 3150    C_ wss 3152   {csn 3640   {cpr 3641   {ctp 3642
This theorem is referenced by:  fr3nr  4571  rngplusg  13257  srngplusg  13265  lmodplusg  13274  algaddg  13279  phlplusg  13289  topgrpplusg  13297  otpstset  13304  odrngplusg  13313  odrngle  13316  prdsplusg  13358  prdsle  13361  imasplusg  13420  imasle  13425  fuchom  13835  setchomfval  13911  catchomfval  13930  xpchomfval  13953  psrplusg  16126  psrvscafval  16135  cnfldadd  16384  cnfldle  16388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-pr 3647  df-tp 3648
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