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Theorem tpcoma 3698
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 3680 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21uneq1i 3297 . 2 ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3 df-tp 3612 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 3612 . 2 {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
52, 3, 43eqtr4i 2218 1 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
Colors of variables: wff set class
Syntax hints:   = wceq 1363  cun 3139  {csn 3604  {cpr 3605  {ctp 3606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-pr 3611  df-tp 3612
This theorem is referenced by:  tpcomb  3699
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