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Theorem snsstp1 4379
Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
Assertion
Ref Expression
snsstp1 {𝐴} ⊆ {𝐴, 𝐵, 𝐶}

Proof of Theorem snsstp1
StepHypRef Expression
1 snsspr1 4377 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
2 ssun1 3809 . . 3 {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
31, 2sstri 3645 . 2 {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 4215 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
53, 4sseqtr4i 3671 1 {𝐴} ⊆ {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  rngbase  16048  srngbase  16056  lmodbase  16065  ipsbase  16072  ipssca  16075  phlbase  16082  topgrpbas  16090  otpsbas  16099  otpsbasOLD  16103  odrngbas  16114  odrngtset  16117  prdssca  16163  prdsbas  16164  prdstset  16173  imasbas  16219  imassca  16226  imastset  16229  fucbas  16667  setcbas  16775  catcbas  16794  estrcbas  16812  xpcbas  16865  psrbas  19426  psrsca  19437  cnfldbas  19798  cnfldtset  19802  trkgbas  25389  signswch  30766  algbase  38065  clsk1indlem4  38659  clsk1indlem1  38660  rngcbasALTV  42308  ringcbasALTV  42371
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