Theorem snsstp2 4380

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 4378	. . 3 ⊢ {𝐵} ⊆ {𝐴, 𝐵}
2		ssun1 3809	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3645	. 2 ⊢ {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4215	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtr4i 3671	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3605   ⊆ wss 3607  {csn 4210  {cpr 4212  {ctp 4214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-pr 4213  df-tp 4215

This theorem is referenced by:  fr3nr  7021  rngplusg  16049  srngplusg  16057  lmodplusg  16066  ipsaddg  16073  ipsvsca  16076  phlplusg  16083  topgrpplusg  16091  otpstset  16100  otpstsetOLD  16104  odrngplusg  16115  odrngle  16118  prdsplusg  16165  prdsvsca  16167  prdsle  16169  imasplusg  16224  imasvsca  16227  imasle  16230  fuchom  16668  setchomfval  16776  catchomfval  16795  estrchomfval  16813  xpchomfval  16866  psrplusg  19429  psrvscafval  19438  cnfldadd  19799  cnfldle  19803  trkgdist  25390  algaddg  38066  clsk1indlem4  38659  rngchomfvalALTV  42309  ringchomfvalALTV  42372