Theorem snsstp1 4816

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 4814	. . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵}
2		ssun1 4178	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3993	. 2 ⊢ {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4631	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtrri 4033	1 ⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3949   ⊆ wss 3951  {csn 4626  {cpr 4628  {ctp 4630

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-pr 4629  df-tp 4631

This theorem is referenced by:  fr3nr  7792  rngbase  17343  srngbase  17354  lmodbase  17370  ipsbase  17381  ipssca  17384  phlbase  17391  topgrpbas  17406  otpsbas  17421  odrngbas  17448  odrngtset  17451  prdssca  17501  prdsbas  17502  prdstset  17511  imasbas  17557  imassca  17564  imastset  17567  fucbas  18008  setcbas  18123  catcbas  18146  estrcbas  18169  cnfldbas  21368  cnfldtset  21374  cnfldbasOLD  21383  cnfldtsetOLD  21387  psrbas  21953  psrsca  21967  trkgbas  28453  rlocbas  33271  rlocaddval  33272  rlocmulval  33273  idlsrgbas  33532  signswch  34576  algbase  43186  clsk1indlem4  44057  clsk1indlem1  44058  cycl3grtri  47914  rngcbasALTV  48182  ringcbasALTV  48216  mndtcbasval  49177