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Theorem snsstp1 3782
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 3780 . . 3  |-  { A }  C_  { A ,  B }
2 ssun1 3351 . . 3  |-  { A ,  B }  C_  ( { A ,  B }  u.  { C } )
31, 2sstri 3201 . 2  |-  { A }  C_  ( { A ,  B }  u.  { C } )
4 df-tp 3661 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
53, 4sseqtr4i 3224 1  |-  { A }  C_  { A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    u. cun 3163    C_ wss 3165   {csn 3653   {cpr 3654   {ctp 3655
This theorem is referenced by:  fr3nr  4587  rngbase  13272  srngbase  13280  lmodbase  13289  algbase  13294  phlbase  13304  topgrpbas  13312  otpsbas  13319  odrngbas  13328  odrngtset  13331  prdsbas  13373  prdstset  13381  imasbas  13431  imastset  13440  fucbas  13850  setcbas  13926  catcbas  13945  xpcbas  13968  psrbas  16140  psrsca  16150  cnfldbas  16399  cnfldtset  16403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-pr 3660  df-tp 3661
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