Theorem tpcoma 4689

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

Step	Hyp	Ref	Expression
1		prcom 4671	. . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	uneq1i 4138	. 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3		df-tp 4575	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4575	. 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
5	2, 3, 4	3eqtr4i 2857	1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}

Syntax hints:   = wceq 1536   ∪ cun 3937  {csn 4570  {cpr 4572  {ctp 4574

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-un 3944  df-pr 4573  df-tp 4575

This theorem is referenced by:  tpcomb  4690  tppreqb  4741  nb3grpr2  27168  nb3gr2nb  27169  frgr3v  28057  3vfriswmgr  28060  1to3vfriswmgr  28062