Theorem tpcoma 4717

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 4699	. . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	uneq1i 4130	. 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3		df-tp 4597	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4597	. 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
5	2, 3, 4	3eqtr4i 2763	1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}

Syntax hints:   = wceq 1540   ∪ cun 3915  {csn 4592  {cpr 4594  {ctp 4596

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3922  df-pr 4595  df-tp 4597

This theorem is referenced by:  tpcomb  4718  tppreqb  4772  nb3grpr2  29317  nb3gr2nb  29318  frgr3v  30211  3vfriswmgr  30214  1to3vfriswmgr  30216