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Theorem snsstp2 4321
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
StepHypRef Expression
1 snsspr2 4319 . . 3 {𝐵} ⊆ {𝐴, 𝐵}
2 ssun1 3759 . . 3 {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
31, 2sstri 3596 . 2 {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 4158 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
53, 4sseqtr4i 3622 1 {𝐵} ⊆ {𝐴, 𝐵, 𝐶}
Colors of variables: wff setvar class
Syntax hints:  cun 3557  wss 3559  {csn 4153  {cpr 4155  {ctp 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-un 3564  df-in 3566  df-ss 3573  df-pr 4156  df-tp 4158
This theorem is referenced by:  fr3nr  6933  rngplusg  15930  srngplusg  15938  lmodplusg  15947  ipsaddg  15954  ipsvsca  15957  phlplusg  15964  topgrpplusg  15972  otpstset  15981  otpstsetOLD  15985  odrngplusg  15996  odrngle  15999  prdsplusg  16046  prdsvsca  16048  prdsle  16050  imasplusg  16105  imasvsca  16108  imasle  16111  fuchom  16549  setchomfval  16657  catchomfval  16676  estrchomfval  16694  xpchomfval  16747  psrplusg  19309  psrvscafval  19318  cnfldadd  19679  cnfldle  19683  trkgdist  25258  algaddg  37257  clsk1indlem4  37851  rngchomfvalALTV  41293  ringchomfvalALTV  41356
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