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Theorem snsstp1 3942
Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
Assertion
Ref Expression
snsstp1  |-  { A }  C_  { A ,  B ,  C }

Proof of Theorem snsstp1
StepHypRef Expression
1 snsspr1 3940 . . 3  |-  { A }  C_  { A ,  B }
2 ssun1 3503 . . 3  |-  { A ,  B }  C_  ( { A ,  B }  u.  { C } )
31, 2sstri 3350 . 2  |-  { A }  C_  ( { A ,  B }  u.  { C } )
4 df-tp 3815 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
53, 4sseqtr4i 3374 1  |-  { A }  C_  { A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    u. cun 3311    C_ wss 3313   {csn 3807   {cpr 3808   {ctp 3809
This theorem is referenced by:  fr3nr  4753  rngbase  13570  srngbase  13578  lmodbase  13587  algbase  13592  phlbase  13602  topgrpbas  13610  otpsbas  13617  odrngbas  13628  odrngtset  13631  prdsbas  13673  prdstset  13681  imasbas  13731  imastset  13740  fucbas  14150  setcbas  14226  catcbas  14245  xpcbas  14268  psrbas  16436  psrsca  16446  cnfldbas  16700  cnfldtset  16704
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2951  df-un 3318  df-in 3320  df-ss 3327  df-pr 3814  df-tp 3815
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