Theorem snsstp3 4788

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 4140	. 2 ⊢ {𝐶} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2		df-tp 4599	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sseqtrri 3994	1 ⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3911   ⊆ wss 3913  {csn 4594  {cpr 4596  {ctp 4598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-tp 4599

This theorem is referenced by:  fr3nr  7771  rngmulr  17354  srngmulr  17365  lmodsca  17381  ipsmulr  17392  ipsip  17395  phlsca  17402  topgrptset  17417  otpsle  17432  odrngmulr  17459  odrngds  17462  prdsmulr  17512  prdsip  17514  prdsds  17517  imasds  17567  imasmulr  17572  imasip  17575  fuccofval  18019  setccofval  18139  catccofval  18161  estrccofval  18185  xpccofval  18238  mpocnfldmul  21498  cnfldds  21503  psrmulr  22061  trkgitv  28682  rlocmulval  33531  idlsrgmulr  33742  signswch  34893  algmulr  43795  clsk1indlem1  44663  rngccofvalALTV  48924  ringccofvalALTV  48958  catcofval  49891  mndtcco  50248