Theorem snsstp2 4817

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 4815	. . 3 ⊢ {𝐵} ⊆ {𝐴, 𝐵}
2		ssun1 4178	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3993	. 2 ⊢ {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4631	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtrri 4033	1 ⊢ {𝐵} ⊆ {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  rngplusg  17344  srngplusg  17355  lmodplusg  17371  ipsaddg  17382  ipsvsca  17385  phlplusg  17392  topgrpplusg  17407  otpstset  17422  odrngplusg  17449  odrngle  17452  prdsplusg  17503  prdsvsca  17505  prdsle  17507  imasplusg  17562  imasvsca  17565  imasle  17568  fuchom  18009  setchomfval  18124  catchomfval  18147  estrchomfval  18170  xpchomfval  18224  mpocnfldadd  21369  cnfldle  21375  cnfldaddOLD  21384  cnfldleOLD  21388  psrplusg  21956  psrvscafval  21968  trkgdist  28454  rlocaddval  33272  idlsrgplusg  33533  algaddg  43187  clsk1indlem4  44057  rngchomfvalALTV  48183  ringchomfvalALTV  48217  mndtchom  49181