Theorem snsstp3 4769

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 4130	. 2 ⊢ {𝐶} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2		df-tp 4582	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sseqtrri 3985	1 ⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3901   ⊆ wss 3903  {csn 4577  {cpr 4579  {ctp 4581

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-un 3908  df-ss 3920  df-tp 4582

This theorem is referenced by:  fr3nr  7708  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  21268  cnfldds  21273  cnfldmulOLD  21282  cnflddsOLD  21286  psrmulr  21849  trkgitv  28392  rlocmulval  33209  idlsrgmulr  33444  signswch  34529  algmulr  43153  clsk1indlem1  44022  rngccofvalALTV  48258  ringccofvalALTV  48292  catcofval  49217  mndtcco  49574