Theorem snsstp3 4794

Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Assertion

Ref	Expression
snsstp3	⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Proof of Theorem snsstp3

Step	Hyp	Ref	Expression
1		ssun2 4154	. 2 ⊢ {𝐶} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2		df-tp 4606	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sseqtrri 4008	1 ⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3924   ⊆ wss 3926  {csn 4601  {cpr 4603  {ctp 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-ss 3943  df-tp 4606

This theorem is referenced by:  fr3nr  7766  rngmulr  17315  srngmulr  17326  lmodsca  17342  ipsmulr  17353  ipsip  17356  phlsca  17363  topgrptset  17378  otpsle  17393  odrngmulr  17420  odrngds  17423  prdsmulr  17473  prdsip  17475  prdsds  17478  imasds  17527  imasmulr  17532  imasip  17535  fuccofval  17975  setccofval  18095  catccofval  18117  estrccofval  18141  xpccofval  18194  mpocnfldmul  21322  cnfldds  21327  cnfldmulOLD  21336  cnflddsOLD  21340  psrmulr  21902  trkgitv  28426  rlocmulval  33264  idlsrgmulr  33522  signswch  34593  algmulr  43200  clsk1indlem1  44069  rngccofvalALTV  48245  ringccofvalALTV  48279  catcofval  49148  mndtcco  49462