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Theorem tpidm12 3622
 Description: Unordered triple {𝐴, 𝐴, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
Assertion
Ref Expression
tpidm12 {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}

Proof of Theorem tpidm12
StepHypRef Expression
1 dfsn2 3541 . . 3 {𝐴} = {𝐴, 𝐴}
21uneq1i 3226 . 2 ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵})
3 df-pr 3534 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 df-tp 3535 . 2 {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵})
52, 3, 43eqtr4ri 2171 1 {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ∪ cun 3069  {csn 3527  {cpr 3528  {ctp 3529 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-pr 3534  df-tp 3535 This theorem is referenced by:  tpidm13  3623  tpidm23  3624  tpidm  3625
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