Theorem tpcoma

Description: Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015)

	Ref	Expression
Assertion	tpcoma	⊢ { 𝐴 , 𝐵 , 𝐶 } = { 𝐵 , 𝐴 , 𝐶 }

Proof

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