Theorem snsstp3 4818

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 4179	. 2 ⊢ {𝐶} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2		df-tp 4631	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sseqtrri 4033	1 ⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3949   ⊆ wss 3951  {csn 4626  {cpr 4628  {ctp 4630

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-tp 4631

This theorem is referenced by:  fr3nr  7792  rngmulr  17345  srngmulr  17356  lmodsca  17372  ipsmulr  17383  ipsip  17386  phlsca  17393  topgrptset  17408  otpsle  17423  odrngmulr  17450  odrngds  17453  prdsmulr  17504  prdsip  17506  prdsds  17509  imasds  17558  imasmulr  17563  imasip  17566  fuccofval  18007  setccofval  18127  catccofval  18149  estrccofval  18173  xpccofval  18227  mpocnfldmul  21371  cnfldds  21376  cnfldmulOLD  21385  cnflddsOLD  21389  psrmulr  21962  trkgitv  28455  rlocmulval  33273  idlsrgmulr  33535  signswch  34576  algmulr  43188  clsk1indlem1  44058  rngccofvalALTV  48186  ringccofvalALTV  48220  mndtcco  49182