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Theorem tpcoma 4750
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 4732 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21uneq1i 4164 . 2 ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3 df-tp 4631 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 4631 . 2 {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
52, 3, 43eqtr4i 2775 1 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cun 3949  {csn 4626  {cpr 4628  {ctp 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-pr 4629  df-tp 4631
This theorem is referenced by:  tpcomb  4751  tppreqb  4805  nb3grpr2  29400  nb3gr2nb  29401  frgr3v  30294  3vfriswmgr  30297  1to3vfriswmgr  30299
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