Theorem unundir

Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004)

	Ref	Expression
Assertion	unundir	⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) = ( ( 𝐴 ∪ 𝐶 ) ∪ ( 𝐵 ∪ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		unidm	⊢ ( 𝐶 ∪ 𝐶 ) = 𝐶
2	1	uneq2i	⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ ( 𝐶 ∪ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 )
3		un4	⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ ( 𝐶 ∪ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐶 ) ∪ ( 𝐵 ∪ 𝐶 ) )
4	2 3	eqtr3i	⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) = ( ( 𝐴 ∪ 𝐶 ) ∪ ( 𝐵 ∪ 𝐶 ) )