Theorem snsstp1 4760

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

Assertion

Ref	Expression
snsstp1	⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶}

Proof of Theorem snsstp1

Step	Hyp	Ref	Expression
1		snsspr1 4758	. . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
2		ssun1 4119	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3932	. 2 ⊢ {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4573	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtrri 3972	1 ⊢ {𝐴} ⊆ {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  7721  rngbase  17257  srngbase  17268  lmodbase  17284  ipsbase  17295  ipssca  17298  phlbase  17305  topgrpbas  17320  otpsbas  17335  odrngbas  17362  odrngtset  17365  prdssca  17414  prdsbas  17415  prdstset  17424  imasbas  17471  imassca  17478  imastset  17481  fucbas  17925  setcbas  18040  catcbas  18063  estrcbas  18086  cnfldbas  21352  cnfldtset  21358  cnfldbasOLD  21367  cnfldtsetOLD  21371  psrbas  21927  psrsca  21940  trkgbas  28531  rlocbas  33347  rlocaddval  33348  rlocmulval  33349  idlsrgbas  33583  signswch  34725  algbase  43626  clsk1indlem4  44495  clsk1indlem1  44496  cycl3grtri  48441  rngcbasALTV  48760  ringcbasALTV  48794  catbas  49719  mndtcbasval  50073