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Theorem tpcoma 4440
Description: Swap 1st and 2nd members of an unordered 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 4422 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21uneq1i 3925 . 2 ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3 df-tp 4339 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 4339 . 2 {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
52, 3, 43eqtr4i 2797 1 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cun 3730  {csn 4334  {cpr 4336  {ctp 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-pr 4337  df-tp 4339
This theorem is referenced by:  tpcomb  4441  tppreqb  4490  nb3grpr2  26564  nb3gr2nb  26565  frgr3v  27513  3vfriswmgr  27516  1to3vfriswmgr  27518
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