ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snsstp3 GIF version

Theorem snsstp3 3584
Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
Assertion
Ref Expression
snsstp3 {𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Proof of Theorem snsstp3
StepHypRef Expression
1 ssun2 3162 . 2 {𝐶} ⊆ ({𝐴, 𝐵} ∪ {𝐶})
2 df-tp 3449 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
31, 2sseqtr4i 3057 1 {𝐶} ⊆ {𝐴, 𝐵, 𝐶}
Colors of variables: wff set class
Syntax hints:  cun 2995  wss 2997  {csn 3441  {cpr 3442  {ctp 3443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-tp 3449
This theorem is referenced by:  sstpr  3596
  Copyright terms: Public domain W3C validator