Theorem tpcoma 4712

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 4694	. . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	uneq1i 4120	. 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3		df-tp 4590	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4590	. 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
5	2, 3, 4	3eqtr4i 2798	1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}

Syntax hints:   = wceq 1563   ∪ cun 3905  {csn 4585  {cpr 4587  {ctp 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3912  df-pr 4588  df-tp 4590

This theorem is referenced by:  tpcomb  4713  tppreqb  4768  nb3grpr2  29642  nb3gr2nb  29643  frgr3v  30535  3vfriswmgr  30538  1to3vfriswmgr  30540