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Theorem tpcoma 4678
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 4660 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21uneq1i 4133 . 2 ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3 df-tp 4564 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4 df-tp 4564 . 2 {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
52, 3, 43eqtr4i 2852 1 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  cun 3932  {csn 4559  {cpr 4561  {ctp 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-un 3939  df-pr 4562  df-tp 4564
This theorem is referenced by:  tpcomb  4679  tppreqb  4730  nb3grpr2  27157  nb3gr2nb  27158  frgr3v  28046  3vfriswmgr  28049  1to3vfriswmgr  28051
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