Theorem snsstp1 4774

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 4772	. . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
2		ssun1 4132	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3945	. 2 ⊢ {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4587	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtrri 3985	1 ⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3901   ⊆ wss 3903  {csn 4582  {cpr 4584  {ctp 4586

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 3444  df-un 3908  df-ss 3920  df-pr 4585  df-tp 4587

This theorem is referenced by:  fr3nr  7729  rngbase  17233  srngbase  17244  lmodbase  17260  ipsbase  17271  ipssca  17274  phlbase  17281  topgrpbas  17296  otpsbas  17311  odrngbas  17338  odrngtset  17341  prdssca  17390  prdsbas  17391  prdstset  17400  imasbas  17447  imassca  17454  imastset  17457  fucbas  17901  setcbas  18016  catcbas  18039  estrcbas  18062  cnfldbas  21330  cnfldtset  21336  cnfldbasOLD  21345  cnfldtsetOLD  21349  psrbas  21906  psrsca  21920  trkgbas  28534  rlocbas  33367  rlocaddval  33368  rlocmulval  33369  idlsrgbas  33603  signswch  34745  algbase  43560  clsk1indlem4  44429  clsk1indlem1  44430  cycl3grtri  48336  rngcbasALTV  48655  ringcbasALTV  48689  catbas  49614  mndtcbasval  49968