Theorem snsstp2 4761

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 4759	. . 3 ⊢ {𝐵} ⊆ {𝐴, 𝐵}
2		ssun1 4119	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3932	. 2 ⊢ {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4573	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtrri 3972	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3888   ⊆ wss 3890  {csn 4568  {cpr 4570  {ctp 4572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-ss 3907  df-pr 4571  df-tp 4573

This theorem is referenced by:  fr3nr  7717  rngplusg  17221  srngplusg  17232  lmodplusg  17248  ipsaddg  17259  ipsvsca  17262  phlplusg  17269  topgrpplusg  17284  otpstset  17299  odrngplusg  17326  odrngle  17329  prdsplusg  17379  prdsvsca  17381  prdsle  17383  imasplusg  17439  imasvsca  17442  imasle  17445  fuchom  17889  setchomfval  18004  catchomfval  18027  estrchomfval  18050  xpchomfval  18103  mpocnfldadd  21316  cnfldle  21322  cnfldaddOLD  21331  cnfldleOLD  21335  psrplusg  21893  psrvscafval  21905  trkgdist  28502  rlocaddval  33334  idlsrgplusg  33570  algaddg  43606  clsk1indlem4  44474  rngchomfvalALTV  48701  ringchomfvalALTV  48735  cathomfval  49660  mndtchom  50017