Theorem snsstp3 4767

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 4126	. 2 ⊢ {𝐶} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2		df-tp 4578	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sseqtrri 3979	1 ⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3895   ⊆ wss 3897  {csn 4573  {cpr 4575  {ctp 4577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-ss 3914  df-tp 4578

This theorem is referenced by:  fr3nr  7705  rngmulr  17205  srngmulr  17216  lmodsca  17232  ipsmulr  17243  ipsip  17246  phlsca  17253  topgrptset  17268  otpsle  17283  odrngmulr  17310  odrngds  17313  prdsmulr  17363  prdsip  17365  prdsds  17368  imasds  17417  imasmulr  17422  imasip  17425  fuccofval  17869  setccofval  17989  catccofval  18011  estrccofval  18035  xpccofval  18088  mpocnfldmul  21298  cnfldds  21303  cnfldmulOLD  21312  cnflddsOLD  21316  psrmulr  21879  trkgitv  28425  rlocmulval  33236  idlsrgmulr  33472  signswch  34574  algmulr  43279  clsk1indlem1  44148  rngccofvalALTV  48380  ringccofvalALTV  48414  catcofval  49339  mndtcco  49696