Metamath Proof Explorer


Theorem unundir

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

Ref Expression
Assertion unundir ( ( 𝐴𝐵 ) ∪ 𝐶 ) = ( ( 𝐴𝐶 ) ∪ ( 𝐵𝐶 ) )

Proof

Step Hyp Ref Expression
1 unidm ( 𝐶𝐶 ) = 𝐶
2 1 uneq2i ( ( 𝐴𝐵 ) ∪ ( 𝐶𝐶 ) ) = ( ( 𝐴𝐵 ) ∪ 𝐶 )
3 un4 ( ( 𝐴𝐵 ) ∪ ( 𝐶𝐶 ) ) = ( ( 𝐴𝐶 ) ∪ ( 𝐵𝐶 ) )
4 2 3 eqtr3i ( ( 𝐴𝐵 ) ∪ 𝐶 ) = ( ( 𝐴𝐶 ) ∪ ( 𝐵𝐶 ) )