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Theorem tpidm12 4667
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 4556 . . 3 {𝐴} = {𝐴, 𝐴}
21uneq1i 4114 . 2 ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵})
3 df-pr 4546 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 df-tp 4548 . 2 {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵})
52, 3, 43eqtr4ri 2854 1 {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cun 3911  {csn 4543  {cpr 4545  {ctp 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-v 3475  df-un 3918  df-pr 4546  df-tp 4548
This theorem is referenced by:  tpidm13  4668  tpidm23  4669  tpidm  4670  fntpb  6948  hashtpg  13828
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