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Theorem tpidm12 3706
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 3621 . . 3 {𝐴} = {𝐴, 𝐴}
21uneq1i 3300 . 2 ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵})
3 df-pr 3614 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 df-tp 3615 . 2 {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵})
52, 3, 43eqtr4ri 2221 1 {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}
Colors of variables: wff set class
Syntax hints:   = wceq 1364  cun 3142  {csn 3607  {cpr 3608  {ctp 3609
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-pr 3614  df-tp 3615
This theorem is referenced by:  tpidm13  3707  tpidm23  3708  tpidm  3709
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