Theorem snsstp2 4793

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

Assertion

Ref	Expression
snsstp2	⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Proof of Theorem snsstp2

Step	Hyp	Ref	Expression
1		snsspr2 4791	. . 3 ⊢ {𝐵} ⊆ {𝐴, 𝐵}
2		ssun1 4153	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3968	. 2 ⊢ {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4606	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtrri 4008	1 ⊢ {𝐵} ⊆ {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-pr 4604  df-tp 4606

This theorem is referenced by:  fr3nr  7764  rngplusg  17312  srngplusg  17323  lmodplusg  17339  ipsaddg  17350  ipsvsca  17353  phlplusg  17360  topgrpplusg  17375  otpstset  17390  odrngplusg  17417  odrngle  17420  prdsplusg  17470  prdsvsca  17472  prdsle  17474  imasplusg  17529  imasvsca  17532  imasle  17535  fuchom  17975  setchomfval  18090  catchomfval  18113  estrchomfval  18136  xpchomfval  18189  mpocnfldadd  21318  cnfldle  21324  cnfldaddOLD  21333  cnfldleOLD  21337  psrplusg  21894  psrvscafval  21906  trkgdist  28371  rlocaddval  33209  idlsrgplusg  33466  algaddg  43146  clsk1indlem4  44015  rngchomfvalALTV  48190  ringchomfvalALTV  48224  cathomfval  49095  mndtchom  49409