MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snsstp2 Structured version   Visualization version   GIF version

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
StepHypRef Expression
1 snsspr2 4378 . . 3 {𝐵} ⊆ {𝐴, 𝐵}
2 ssun1 3809 . . 3 {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
31, 2sstri 3645 . 2 {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 4215 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
53, 4sseqtr4i 3671 1 {𝐵} ⊆ {𝐴, 𝐵, 𝐶}
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator