Theorem qdass

Description: Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016)

	Ref	Expression
Assertion	qdass	⊢ ( { 𝐴 , 𝐵 } ∪ { 𝐶 , 𝐷 } ) = ( { 𝐴 , 𝐵 , 𝐶 } ∪ { 𝐷 } )

Proof

Step	Hyp	Ref	Expression
1		unass	⊢ ( ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) ∪ { 𝐷 } ) = ( { 𝐴 , 𝐵 } ∪ ( { 𝐶 } ∪ { 𝐷 } ) )
2		df-tp	⊢ { 𝐴 , 𝐵 , 𝐶 } = ( { 𝐴 , 𝐵 } ∪ { 𝐶 } )
3	2	uneq1i	⊢ ( { 𝐴 , 𝐵 , 𝐶 } ∪ { 𝐷 } ) = ( ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) ∪ { 𝐷 } )
4		df-pr	⊢ { 𝐶 , 𝐷 } = ( { 𝐶 } ∪ { 𝐷 } )
5	4	uneq2i	⊢ ( { 𝐴 , 𝐵 } ∪ { 𝐶 , 𝐷 } ) = ( { 𝐴 , 𝐵 } ∪ ( { 𝐶 } ∪ { 𝐷 } ) )
6	1 3 5	3eqtr4ri	⊢ ( { 𝐴 , 𝐵 } ∪ { 𝐶 , 𝐷 } ) = ( { 𝐴 , 𝐵 , 𝐶 } ∪ { 𝐷 } )