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Theorem snsstp1 3539
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 3537 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
2 ssun1 3131 . . 3 {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
31, 2sstri 2979 . 2 {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 3408 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
53, 4sseqtr4i 3003 1 {𝐴} ⊆ {𝐴, 𝐵, 𝐶}
Colors of variables: wff set class
Syntax hints:  cun 2940  wss 2942  {csn 3400  {cpr 3401  {ctp 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pr 3407  df-tp 3408
This theorem is referenced by: (None)
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