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Theorem tpidm12 4426
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 4329 . . 3 {𝐴} = {𝐴, 𝐴}
21uneq1i 3914 . 2 ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵})
3 df-pr 4319 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
4 df-tp 4321 . 2 {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵})
52, 3, 43eqtr4ri 2804 1 {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cun 3721  {csn 4316  {cpr 4318  {ctp 4320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728  df-pr 4319  df-tp 4321
This theorem is referenced by:  tpidm13  4427  tpidm23  4428  tpidm  4429  fntpb  6615  hashtpg  13462
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