Theorem tpcoma 4711

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 4693	. . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	uneq1i 4111	. 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶}) = ({𝐵, 𝐴} ∪ {𝐶})
3		df-tp 4589	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
4		df-tp 4589	. 2 ⊢ {𝐵, 𝐴, 𝐶} = ({𝐵, 𝐴} ∪ {𝐶})
5	2, 3, 4	3eqtr4i 2794	1 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3897  {csn 4584  {cpr 4586  {ctp 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-pr 4587  df-tp 4589

This theorem is used by:  tpcomb  4712  tppreqb  4768  nb3grpr2  29946  nb3gr2nb  29947  frgr3v  30858  3vfriswmgr  30861  1to3vfriswmgr  30863