Theorem tpidm12

Description: Unordered triple { A , A , B } is just an overlong way to write { A , B } . (Contributed by David A. Wheeler, 10-May-2015)

	Ref	Expression
Assertion	tpidm12	⊢ { 𝐴 , 𝐴 , 𝐵 } = { 𝐴 , 𝐵 }

Proof

Step	Hyp	Ref	Expression
1		dfsn2	⊢ { 𝐴 } = { 𝐴 , 𝐴 }
2	1	uneq1i	⊢ ( { 𝐴 } ∪ { 𝐵 } ) = ( { 𝐴 , 𝐴 } ∪ { 𝐵 } )
3		df-pr	⊢ { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )
4		df-tp	⊢ { 𝐴 , 𝐴 , 𝐵 } = ( { 𝐴 , 𝐴 } ∪ { 𝐵 } )
5	2 3 4	3eqtr4ri	⊢ { 𝐴 , 𝐴 , 𝐵 } = { 𝐴 , 𝐵 }