Theorem snsstp2 4744

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 4742	. . 3 ⊢ {𝐵} ⊆ {𝐴, 𝐵}
2		ssun1 4148	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3976	. 2 ⊢ {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4566	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtrri 4004	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3934   ⊆ wss 3936  {csn 4561  {cpr 4563  {ctp 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-un 3941  df-in 3943  df-ss 3952  df-pr 4564  df-tp 4566

This theorem is referenced by:  fr3nr  7488  rngplusg  16615  srngplusg  16623  lmodplusg  16632  ipsaddg  16639  ipsvsca  16642  phlplusg  16649  topgrpplusg  16657  otpstset  16664  odrngplusg  16675  odrngle  16678  prdsplusg  16725  prdsvsca  16727  prdsle  16729  imasplusg  16784  imasvsca  16787  imasle  16790  fuchom  17225  setchomfval  17333  catchomfval  17352  estrchomfval  17370  xpchomfval  17423  psrplusg  20155  psrvscafval  20164  cnfldadd  20544  cnfldle  20548  trkgdist  26226  algaddg  39772  clsk1indlem4  40387  rngchomfvalALTV  44248  ringchomfvalALTV  44311