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

Theorem tpcoma 3626
 Description: Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
Assertion
Ref Expression
tpcoma {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}

Proof of Theorem tpcoma
StepHypRef Expression
1 prcom 3608 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21uneq1i 3232 . 2 ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3 df-tp 3541 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 3541 . 2 {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
52, 3, 43eqtr4i 2171 1 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∪ cun 3075  {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-pr 3540  df-tp 3541 This theorem is referenced by:  tpcomb  3627
 Copyright terms: Public domain W3C validator