Theorem tpcoma 4755

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 4737	. . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	uneq1i 4174	. 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3		df-tp 4636	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4636	. 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
5	2, 3, 4	3eqtr4i 2773	1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}

Syntax hints:   = wceq 1537   ∪ cun 3961  {csn 4631  {cpr 4633  {ctp 4635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-pr 4634  df-tp 4636

This theorem is referenced by:  tpcomb  4756  tppreqb  4810  nb3grpr2  29415  nb3gr2nb  29416  frgr3v  30304  3vfriswmgr  30307  1to3vfriswmgr  30309