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Theorem snsstp1 3679
 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 3677 . . 3 {𝐴} ⊆ {𝐴, 𝐵}
2 ssun1 3245 . . 3 {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
31, 2sstri 3112 . 2 {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 3541 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
53, 4sseqtrri 3138 1 {𝐴} ⊆ {𝐴, 𝐵, 𝐶}
 Colors of variables: wff set class Syntax hints:   ∪ cun 3075   ⊆ wss 3077  {csn 3533  {cpr 3534  {ctp 3535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-pr 3540  df-tp 3541 This theorem is referenced by: (None)
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