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Theorem snsstp1 4501
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 4499 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
2 ssun1 3938 . . 3 {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
31, 2sstri 3770 . 2 {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 4339 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
53, 4sseqtr4i 3798 1 {𝐴} ⊆ {𝐴, 𝐵, 𝐶}
Colors of variables: wff setvar class
Syntax hints:  cun 3730  wss 3732  {csn 4334  {cpr 4336  {ctp 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-in 3739  df-ss 3746  df-pr 4337  df-tp 4339
This theorem is referenced by:  fr3nr  7177  rngbase  16275  srngbase  16283  lmodbase  16292  ipsbase  16299  ipssca  16302  phlbase  16309  topgrpbas  16317  otpsbas  16324  odrngbas  16335  odrngtset  16338  prdssca  16384  prdsbas  16385  prdstset  16394  imasbas  16440  imassca  16447  imastset  16450  fucbas  16887  setcbas  16995  catcbas  17014  estrcbas  17032  xpcbas  17086  psrbas  19652  psrsca  19663  cnfldbas  20023  cnfldtset  20027  trkgbas  25635  signswch  31021  algbase  38357  clsk1indlem4  38948  clsk1indlem1  38949  rngcbasALTV  42584  ringcbasALTV  42647
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