Theorem prcom

Description: Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993)

	Ref	Expression
Assertion	prcom	⊢ { 𝐴 , 𝐵 } = { 𝐵 , 𝐴 }

Proof

Step	Hyp	Ref	Expression
1		uncom	⊢ ( { 𝐴 } ∪ { 𝐵 } ) = ( { 𝐵 } ∪ { 𝐴 } )
2		df-pr	⊢ { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )
3		df-pr	⊢ { 𝐵 , 𝐴 } = ( { 𝐵 } ∪ { 𝐴 } )
4	1 2 3	3eqtr4i	⊢ { 𝐴 , 𝐵 } = { 𝐵 , 𝐴 }