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Theorem snsstp1 3766
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 3764 . . 3  |-  { A }  C_  { A ,  B }
2 ssun1 3338 . . 3  |-  { A ,  B }  C_  ( { A ,  B }  u.  { C } )
31, 2sstri 3188 . 2  |-  { A }  C_  ( { A ,  B }  u.  { C } )
4 df-tp 3648 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
53, 4sseqtr4i 3211 1  |-  { A }  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  rngbase  13256  srngbase  13264  lmodbase  13273  algbase  13278  phlbase  13288  topgrpbas  13296  otpsbas  13303  odrngbas  13312  odrngtset  13315  prdsbas  13357  prdstset  13365  imasbas  13415  imastset  13424  fucbas  13834  setcbas  13910  catcbas  13929  xpcbas  13952  psrbas  16124  psrsca  16134  cnfldbas  16383  cnfldtset  16387
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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