Theorem snsstp2 4819

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 4817	. . 3 ⊢ {𝐵} ⊆ {𝐴, 𝐵}
2		ssun1 4171	. . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
3	1, 2	sstri 3990	. 2 ⊢ {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4632	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
5	3, 4	sseqtrri 4018	1 ⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Syntax hints:   ∪ cun 3945   ⊆ wss 3947  {csn 4627  {cpr 4629  {ctp 4631

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-un 3952  df-in 3954  df-ss 3964  df-pr 4630  df-tp 4632

This theorem is referenced by:  fr3nr  7761  rngplusg  17249  srngplusg  17260  lmodplusg  17276  ipsaddg  17287  ipsvsca  17290  phlplusg  17297  topgrpplusg  17312  otpstset  17327  odrngplusg  17354  odrngle  17357  prdsplusg  17408  prdsvsca  17410  prdsle  17412  imasplusg  17467  imasvsca  17470  imasle  17473  fuchom  17917  fuchomOLD  17918  setchomfval  18033  catchomfval  18056  estrchomfval  18081  xpchomfval  18135  cnfldadd  21149  cnfldle  21153  psrplusg  21719  psrvscafval  21728  trkgdist  27964  idlsrgplusg  32893  mpocnfldadd  35476  gg-cnfldle  35480  algaddg  42223  clsk1indlem4  43097  rngchomfvalALTV  46970  ringchomfvalALTV  47033  mndtchom  47797